Interesting guesses, but I believe you all managed to miss the non-Air-Force-service in question.
I expect and demand your very best. Anything less, you should have joined the Air Force.
Captain Ramsey, under operating procedures governing the release of nuclear weapons we cannot launch our missiles unless both you and I agree.
Not to shoot you down but FYI:
GI Jane = Navy (Seals to be exact)
Private Benjamin = Army
Full metal Jacket = Marines
NONE of these refer to the Air Force, therefore i believe WE all managed to understand the quote.
Allright, I know this one I just can't place it right now... It's one of 2. I go with Crimson Tide