There are lounges for serving military in many American airports, not just JFK, even some small ones near the major facilities have them. I am not sure what the reaction would be for a non US soldier turning up at the door, but I'll hazard a guess that it will be more than welcoming. These lounges are nearer the AA model (and below) than the Concorde Room however! I am not sure taking one's ID on holiday is a good idea in terms of it getting lost (and the ensuing hassle) but then it's possible that US Customs Border Protection may ask for proof of employment - as they would do for anyone entering the States, particularly for the first time. However there are other sorts of proof which could be used. There is a flag on their computer which indicates someone is employed by Allied Forces.

Indeed the US takes a very, very different approach to advertising their men and women serving in their Armed Forces (perhaps a bit over the top) but the UK is sliding gently in this direction too, having been too long in the category you describe. When Northern Ireland was going on you could understand why, less so now. It's not so unusual to see arriving soldiers in uniform at EDI in particular and to some extent T5.