Was Shakespeare ever a soldier?

I recently read Quartered Safe Out Here by George MacDonald Fraser, which was about his wartime experiences in Burma. He wrote the Flashman books and is one of my favourite authors. In one chapter he recounts how an NCO borrowed his copy of Henry V, and was surprised to find that he liked it, and what's more, thought that Shakespeare must have been in the army himself. It's such a good bit, I typed it out.

I was lying on my groundsheet, renewing acquaintance with Jerome and the tin of pineapples, when Sergeant Hutton squatted down beside me.
“W'at ye readin', then? W'at's this? 'Enry Vee – bloody 'ell, by William Shekspeer!” He gave me a withering look, and leafed over a page. “Enter Chorus. O for a muse of fire that wad...Fook me!” He riffled the pages. “Aye, weel, we'll 'ev a look.” And such is the way of sergeants, he removed it without by-your-leave' that's one that won't be away long, I thought.
I was wrong. Three days later it had not been returned. And having exhausted Jerome and the magazines I was making do with the Fourteenth Army newspaper, SEAC, famous for its cartoon character, Professor Flitt, a jungle infantryman who commented memorably on the passing scene. And I was reading a verse by the paper's film critic

I really do not care a heck
For handsome Mr Gregory Peck
But I would knock off work at four
To see Miss Dorothy Lamour

when Hutton loafed up and tossed Henry V down beside me and seated himself on the section grub box. A silence followed and I asked him if he had liked it. He indicated the book.
“Was Shekspeer ivver in th' Army?”
I said that most scholars thought not, but that there were blanks in his life, so it was possible that, like his friend Ben Johnson, he had served in the Low Countries, or even in Italy. Hutton shook his head.
“If 'e wesn't in th' Army, Ah'll stand tappin'. 'E knaws too bloody much aboot it, man.”
This was fascinating. Hutton was a military hard case who had probably left school long before 14, and his speech and manner suggested that his normal and infrequent reading consisted of company orders and the sports headlines. But Shakespeare had talked to him across the centuries – admittedly on his own subject. I suggested hesitantly that the Bard might have picked up a good deal just from talking to military men; Hutton brushed the notion aside.
“Nivver! Ye knaw them three – Bates, an' them, talkin' afore the battle? Ye doan't git that frae lissening in pubs, son. Naw, 'e's bin theer.” He gave me the hard, aggressive stare of the Cumbrian who is not to be contradicted. “That's my opinion, any roads. An' them oothers – the Frenchmen, the nawblemen, tryin' to kid on that they couldn't care less, w'en they're ****tin' blue lights? Girraway! An' the Constable tekkin' the piss oot o' watsisname -”
“The Dauphin”
“Aye.” He shook his head in admiration. “Naw, ye've 'eerd it a' afore – in different wurrds, like. Them fower officers, the Englishman an' the Scotsman an' the Irishman an' the Welshman – Ah mean, 'e's got their chat off, 'esn't 'e? Ye could tell w'ich wes w'ich, widoot bein' told. That Welsh booger!” He laughed aloud, a thing he rarely did. “Talk till the bloody coos coom yam, the Taffies!” He frowned. “Naw, Ah nivver rid owt be Shekspeer afore – Ah mean, ye 'ear the name, like...” He shrugged eloquently. “Mind, there's times Ah doan't knaw w'at th' 'ell 'e's talkin' aboot -“
“You and me both,” I said, wondering uneasily if there were more passages obscure to me than there were to him. He sat for a moment and then misquoted (and I am not sure that Shakespeare's version was better):
“There's nut many dies weel that dies in a battle. By Christ, 'e's right theer. It's a good bit, that.” He got up.
“Thanks for the lend on't, Jock”