No. In the concise words of 18th century military author Bennett Cuthbertson, "A Soldier must never be allowed to wear a wig." As /u/Sapientiam stated below, the queue was a popular military hairstyle. However, it was not the style seen predominantly during the Revolution. Other fashions occurred concurrently, and with more frequency, than the queue- Namely the plait and the club. The queue itself appears to have been worn largely by officers.^1 The more common fashions of enlisted men- likely, as Cuthbertson says, because it was a significant economy on ribbon and that it "greases the back of the coat."^2

Military author Bennett Cuthbertson described the 'plait' thusly:

"The hair of the Non-commission-officers, Drummers, and private Men, look tightest, when turned up behind on a comb, and loosely platted, with a black ribband or tape (three quarters long) in a bow knot at the tye."

Only in certain instances did Cuthbertson recommend the use of "artificial aids to elegance":

"when all these aids are not sufficient, from the natural thinness of the hair, a false plat must be added, which, if properly fixt on, can never be discovered: this method is also to be pursued, when a Soldier's hair is but barely long enough to tye, as it will contribute much to the uniform appearance of a Battalion, particularly after having received any number of Recruits."^3

Examples of the plait may be seen on several contemporary illustrations. Nascent forms are present in the paintings of the Swiss David Morier, now present in the Royal Collection. The soldier on the far left on this particular painting wears his hair plaited. A more developed form of the hairstyle is worn by the Grenadiers painted by Philip Jacques de Loutherbourg in 1777. Here, here, and here. The plait is also well illustrated in the famous painting of the death of Major Pierson by J.S. Copley. Again, it is worn by grenadiers.

The Club hairstyle was pulled behind and folded in upon itself and tied with ribbon. I put together a picture of how to wear a cocked hat in the 70's and 80's- Which also includes a number of images of soldiers wearing their hair clubbed. Here it is Clubbed hair is also worn by this light infantryman by Loutherbourg. This style was made regulation for all infantry on July 27 1776- Though Grenadiers, and apparently companies of light infantry, wore their hair plaited. However, Stephen Gilbert writes that the order was not in time to prevent General Howe's army (then campaigning in New York) from cropping their hair short.^4 Such a method of fashioning the hair had been known before. In his widely read Memoirs, a 1740's treatise on soldieirng, Marshal Saxe recommended cropping the soldiers' hair.^5 George Howe, Sir William Howe's late brother, had his hair cropped short, and ordered his entire 55th regiment of light infantry to do the same.^6 Returning to 1776, in contrast to Howe's round heads, the army in Canada under General Burgoyne retained their clubs, as evidenced by the series of contemporary watercolors produced by a German officer serving with the army.^7

Forewarning- I have not watched Turn. However, from what I have seen of the uniforms in the show, the costumers did an abysmal job recreating the look of the British Army in 1776.

Addendum: The 18th Century Material Culture Research Center just dropped this Fantastic Slide Show which contains a wealth of contemporary information on how specifically to maintain the soldier's hair, as well as how to maintain a soldierly appearance in general.

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^1 Thomas Simes, a military author, wrote in 1776: "An Officer, when dressed for Guard, should have his hair queued... The Officers [on field days[, should have their hair queued... The Non-Comissioned Officers and private men must plat (pait) and tuck their hair..." (Quoted in Stephen Gilbert, "The Long and Short of It: Military Hairstyles during the American Revolution Part 1 (of 3): British Army Hairstyles," The Brigade Dispatch XXXV, 15, 2015, 13.) A few contemporary images show men wearing queues (the most famous one being a watercolor by E. Fitzgerald showing a soldier of the Irish artillery). However, the majority of depictions prior to 1776 show the soldiers with their hair either plaited or clubbed, even when their officers wear theirs queued.

^2 Bennett Cutbertson, System for the Complete Interior Economy and Maintenance of a Battalion of Infantry, (London, 1776), 110

^3 Cuthberson, 110.

^4 Gilbert, 15.

^5 Gilbert, 11.

^6 Ibid.

^7 Gilbert, 17. See also the watercolors which I posted on my blog here. /shameless self promotion

Wigs not so much, but powdered hair most definitely, and its worse than you think.

I highly recommend British Military Spectacle: From the Napoleonic Wars through the Crimea by Scott Myerly for a look at some of the zany things that the Brits did. He focuses on the the very latest part of the period when form trumped function in British Military Uniforms but he includes lots of background.

Here's what he says about hair powder:

Hair was worn in a queue, which was coated with a noxious mixture of grease (sometimes rancid) or tallow and then dusted with flour. This made a neat white appearance, especially from a distance, but was a filthy encumbrance to the wearer and apt to attract vermin, especially in extremely overcrowded and often dirty barracks. It is ironic that at a time when soldiers were not fed enough to fill their stomachs, they had to buy powdering flour out of their meager pay. In the Tenth Light Dragoons, officers were required to use a pound of power each week. Most soldiers welcomed the abolition of powdering in 1808 but, typically, the guards kept it longer.

There's another affectation that isn't as well know but is equally pointless: pipeclay. It's the stuff that made those cross-belts so sparkly white.

Pipeclay was another source of superficial (but unhealthy) cleanliness that caused much bother. All white surfaces in the kit--trousers, facings (when white), button laces, drill order jacket, pipping, and gaiters--were painted with pipeclay, which until dry, made it difficult for soldiers to move about, since he had to take care not to brush against anything; this was especially annoying in an overcrowded barracks. The uniform's fabric was damaged by heavy, repeated application of pipeclay, which could also come off in clouds of dust when a regiment was on the march.

edited for spelling

Adding on to what has already been well covered in reference to wigs being worn by the military, hair care in the 18th century for everyone consisted of pomade and powder. Now, it's not the obscene amount of powder which turns the hair completely white that we're talking about. Think of it more like a deep conditioner and a dry shampoo.

Pomatum is based in animal fat, very often sheep or cow. Mixed in with it are other useful items like clove oil (natural insect repellant), lemon, or other essential oils. Having worked with a recipe that was reproduced from Toilette de Flora it is often noted that it smells like Christmas. Applying this to the hair does leave it in a wetter state, which the powder takes care of. It is based in starch, often having other mix ins like oris root (lilies) and bone meal. The powder is applied in the necessary amount to bring the hair texture back to a normal level. This process leaves the hair extremely full and easy to manipulate (it also smells really good). All of this is combed through the hair, combing being considered the necessary way to keep the hair clean. You get down to the roots to prevent any build up. Re-pomading and powdering the hair occurs every couple of weeks.

The hair is also generally kept covered, particularly in work situations, by both men and women. Preventing it from gathering dust, dirt, and soot to begin with makes a huge difference. Women, of course, would wear a cap when they have not dressed their hair up. Men would also wear caps, though of a style more akin to our modern slouch style hats, beanies, or even turbans. Plaiting (braiding) the queue, wrapping it in ribbon, or using a queue bag would do a similar service for the soldier of keeping it clean, untangled, and out of the way.