Regular team colors with camo would be okay, but every branch wears their own pattern now, so you'd be favoring one over the others. Maybe the US flag on the side like the old 9/11 hats would be best? Or just an "Armed Forces Day" patch?
Ironically, the only time I've really liked the way camo look inside a logo is a hat Turtle wears in an old Entourage episode:
Been trying to find that hat for years, it doesn't exist as far as I can tell.