The first step in understanding why this statement is made is to know that every where your Bible uses the word "cross," their Bible uses the word "stake." To confirm their position they will use partial quotes and references from scholars that seem to back up their claim that the Greek word stauros in the New Testament means "stake" or "pole" instead of its true meaning "cross." They also will say, "stauros" in both the classical Greek and Koine carries no thought of a "cross" made of two timbers, but instead it carries the notion of only an upright stake, a pale, pile, or pole.When the Greek lexicons are checked, however, one finds this is not the case.

Heinz: Are JW's using partial quotes? And do Greek Lexicons and dictionaries agree more with Mark's point of view? Let us take a look. "The Tau was the basis for what is now called the "cross" taken from the Latin "crux".  "The shape of the [two-beamed cross] had its origin in ancient Chaldea, and was used as the symbol of the god Tammuz (being in the shape of the mystic Tau, the initial of his name) in that country and in adjacent lands, including Egypt. By the middle of the 3rd cent. A.D. the churches had either departed from, or had travestied, certain doctrines of the Christian faith. In order to increase the prestige of the apostate ecclesiastical system pagans were received into the churches apart from regeneration by faith, and were permitted largely to retain their pagan signs and symbols. Hence the Tau or T, in its most frequent form, with the cross-piece lowered, was adopted to stand for the cross of Christ."—An Expository Dictionary of New Testament Words (London, 1962), W. E. Vine, p. 256.
What is this? The Cross used among ancient pagan? Is there more?
"It is strange, yet unquestionably a fact, that in ages long before the birth of Christ, and since then in lands untouched by the teaching of the Church, the Cross has been used as a sacred symbol. . . . The Greek Bacchus, the Tyrian Tammuz, the Chaldean Bel, and the Norse Odin, were all symbolized to their votaries by a cruciform device."—The Cross in Ritual, Architecture, and Art (London, 1900), G. S. Tyack, p. 1.

The people of the ancient lands used the cross in worship, some, like the Egyptians used it in Phallus worship, or, worship of the male sex organ.  It was used as a symbol of fertility. "Various figures of crosses are found everywhere on Egyptian monuments and tombs, and are considered by many authorities as symbolical either of the phallus [a representation of the male sex organ] or of coition. . . . In Egyptian tombs the crux ansata [cross with a circle or handle on top] is found side by side with the phallus."—A Short History of Sex-Worship (London, 1940), H. Cutner, pp. 16, 17; see also The Non-Christian Cross, p. 183.

The Ancient Church by clergyman W.  D. Killen says, on page 316: "From the most remote antiquity the cross was venerated in Egypt and Syria; it was held in equal honour by the Buddhists of the East; and, what is still more extraordinary, when the Spaniards first visited America, the well-known sign was found among the objects of worship in the idol temples of Anahuac. It is also remarkable that, about the commencement of our era, the pagans were wont to make the sign of a cross upon the forehead in the celebration of some of their sacred mysteries."  The origin of the cross is indeed very pagan.
So you see, the reason that JW's do not use the word bears heavily on its pagan origin. After all, "What accord has Christ with Belial? Or what has a believer in common with an unbeliever? What agreement has the temple of God with idols?" 2Cor 6:15 RSV

"They also will say, "stauros" in both the classical Greek and Koine carries no thought of a "cross" made of two timbers, but instead it carries the notion of only an upright stake, a pale, pile, or pole.
When the Greek lexicons are checked, however, one finds this is not the case. "

We have already checked Vine's, but are there others?
A Comprehensive Dictionary of the Original Greek Words with their Precise Meanings for English
Readers states: "STAUROS . . . denotes, primarily, an upright pale or stake. On such
malefactors were nailed for execution." Similarly, the book The Non-Christian Cross observes:
"There is not a single sentence in any of the numerous writings forming the New Testament,
which, in the original Greek, bears even indirect evidence to the effect that the stauros used in
the case of Jesus was other than an ordinary stauros [pole or stake]; much less to the effect that
it consisted, not of one piece of timber, but of two pieces nailed together in the form of a cross."
Paul Wilhelm Schmidt, who was a professor at the University of Basel, in his work Die
Geschichte Jesu (The History of Jesus), Vol. 2, Tübingen and Leipzig, 1904, pp. 386-394, made
a detailed study of the Greek word stau·ros'. On p. 386 of his work he said: "staur¬V [stau·ros']
means every upright standing pale or tree trunk."
New Bible Dictionary of 1985 under "Cross," page 253: "The Gk. word for 'cross' (stauros; verb
stauroo . . . ) means primarily an upright stake or beam, and secondarily a stake used as an
instrument for punishment and execution."
W. E. Vine says on this subject: "STAUROS (staur¬V) denotes, primarily, an upright pale or
stake. On such malefactors were nailed for execution. Both the noun and the verb stauroo, to
fasten to a stake or pale, are originally to be distinguished from the ecclesiastical form of a two
beamed cross." Greek scholar Vine then mentions the Chaldean origin of the two-piece cross
and how it was adopted from the pagans by Christendom in the third century C.E. as a symbol of
Christ's impalement.—Vine's Expository Dictionary of Old and New Testament Words, 1981, Vol.
1, p. 256.
The Latin dictionary by Lewis and Short gives as the basic meaning of crux "a tree, frame, or
other wooden instruments of execution, on which criminals were impaled or hanged."
The book Dual Heritage—The Bible and the British Museum states: "It may come as a shock to
know that there is no word such as 'cross' in the Greek of the New Testament. The word
translated 'cross' is always the Greek word [stau·ros'] meaning a 'stake' or 'upright pale.' The
cross was not originally a Christian symbol; it is derived from Egypt and Constantine."
See also Strongs and Young's Analytical Concordance.

The Watchtower Society not only claims that Christ did not die on a cross, they further state that there is no evidence that a cross with a crossbeam was ever even used by Romans during the first century. They claim the stake was "the then customary usage of this means of execution in the Orient." They maintain, "The evidence is, therefore, completely lacking that Jesus Christ was crucified on two pieces of timber placed at right angles...The passing of time and further archaeological discoveries will be certain to prove its [torture stake] correctness.

To further elucidate those that see this rejection of a 2-beamed stauros are not WT-born are the follwing references:
"THE sign of the cross has been a symbol of great antiquity, present in nearly every known
culture. Its meaning has eluded anthropologists, though its use in funerary art could well point to
a defense against evil. On the other hand, the famous crux ansata of Egypt, depicted coming
from the mouth, must refer to life or breath. The universal use of the sign of the cross makes
more poignant the striking lack of crosses in early Christian remains, especially any specific
reference to the event on Golgotha. Most scholars now agree that the cross, as an artistic
reference to the passion event, cannot be found prior to the time of Constantine."—Ante
Pacem—Archaeological Evidence of Church Life Before Constantine (1985), by Professor
Graydon F. Snyder, page 27.
"There was no use of the crucifix," says one historian of the early Christians, "and no material
representation of the cross." History of the Christian Church, J. F. Hurst, Vol. I, p. 366.

We have strong Biblical basis for assuming a cross beam:
NAS John 20:25 The other disciples therefore were saying to him, "We have seen the Lord!" But he said to them, "Unless I shall see in His hands the imprint of the nails [plural], and put my finger into the place of the nails [plural], and put my hand into His side, I will not believe.
You will notice in this passage the word "nails," this is in the plural suggesting each hand was nailed seperately to a cross beam. You might also notice in JW literature images of Jesus hanging on a cross with one (singular) nail through his wrist/hands.

The Cyclopaedia of Biblical, Theological, and Ecclesiastical Literature, by M'Clintock and Strong, comments:
'Much time and trouble have been wasted in disputing as to whether three or four nails were used in fastening the Lord. Nonnus affirms that three only were used, in which he is followed by Gregory Nazianzen. The more general belief gives four nails, an opinion which is supported at much length and by curious arguments by Curtius. Others have carried the number of nails as high as fourteen.'-Volume II, page 580.
Accounts of Jesus impalement/crucifixion like Matthew 27:35 give little evidence of the methods used. After Jesus' resurrection, Thomas said: "Unless I see in his hands the print of the nails and stick my finger into the print of the nails and stick my hand into his side, I will certainly not believe." (John 20:25) Because of this some have also concluded from John 20:25 that two nails were used, one through each hand. But does Thomas' use of the plural *nails* have to be understood that Jesus' hands were pierced by a separate nail?
In Luke 24:39 the resurrected Jesus said: "See my hands and my feet, that it is I myself." This suggests that Christ's feet also were nailed. Since Thomas made no mention of nailprints in Jesus' feet, his use of the plural "nails" could have been a general reference to multiple nails used in piercing Jesus.
Debate over such an insignificant detail should not be permitted to becloud the all-important truth that "we were reconciled to God by the death of his Son."-Romans 5:10. RSV

We must also remember too that the Cross was adopted as a "Christian" symbol (312 C.E.)  after it was adopted by Emperor Constantine, who continued to be a Sun-worshipper, and the cross was the symbol of the sun-god, Sol.
January 13, 313 C.E. Constantine as pagan Pontifex Maximus publishes his famous edict of toleration in favor of the professed Christians and they are made eligible to public office.
321 C.E. Sunday Dies Solis, the day of the sun-god, Sol, whose symbol is the cross, is made a day exempt from being judicial and its observance made a legal duty.
325 C.E. Constantine becomes head of the eastern and western sections of the Roman Empire. He calls a religious council for settling the controversy over the Greek characters or "trinity," which threatens the unity of his empire. As pagan Pontifex Maximus, not yet baptized as a Christian, Constantine presides over the council...and the rest is the history of Christendom.
 I think what JW's are driving at is the adoration of a symbol. I think you will agree that many use the figure of a cross a object of worship, and where that happens we are no better than the pagans who did the very same thing. The cross is a symbol of Christ's humiliation, but we are saved through his BLOOD that was shed that day. Let us envision it this way. If our Saviour was killed by a gun, would we be hanging that around our necks. It is too macabre to sprinkle blood on our shirts to commemorate this important event, and using the cross as a symbol of our affection in the Lord Jesus Christ is not necessary. After all, we are walking by faith, not by sight.

Addendum: Here is an interesting note from a recently released Bible:

"The Greek term signifies an upright stake on which criminal were executed, with no suggestion of a cross-beam. In the Latin versions the term 'crux' was used, but according to Livy of the 1st century B.C., the word meant no more than an upright stake; it was only later that crux came to mean a cross. Josephus relates how 2,000 were crucified at one time ('Antiquities' book 17; 10:10) hardly practicable if crosses had to be made for each one. There are Greek words which denote a cross, but none of these appear in the in any of the four gospel accounts of Jesus' execution. At Galatians 3:13 Paul refers to the instrument as 'a timber' (A.V. a tree) a reference to the upright stake on which bodies of criminals were hanged under the Mosaic Law (Deut 21:22), and which Jesus fulfilled by his death.
Some have contended that the Romans did use crosses for execution at that time although Livy refutes this. Even if this were so, the Romans were also careful to observe local customs as fas as possible to avoid unnecessarily upsetting the populace, and so likely would have modified their method to conform to the Jewish practice. A rough upright stake would be in any case less trouble to produce than a hewn cross with a joint strong enough to bear the weight of a man.
Christians are sometimes disturbed to learn that the cross, considered for centuries as a Christian symbol, had its origin long before Christ and was actually used in pagan mythology.It was the symbol of the god Tammuz, and Bacchus, and the Egyptian Osiris. It was worshipped by the Celtic druids and worn on necklaces by the Vestal Virgins of Rome...As the Greek text shows, Christ was not executed on a Cross, that symbol can be regarded for what it is, a pagan corruption of Christian worship introduced in the early centuries of our common era. Thus in harmony with 2 Cor 6:15 although long cherished, it is something that Christians should shun."

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