Federal law....all US military officers MUST be citizens (naturalized OK).
Except for army helos, all US military flying jobs are reserved for commissioned officers (I saw an ALNAV indicating that the navy is experimenting with NCO pilots right now, but you would have to serve as an enlisted sailor before you could apply).
Many other jobs are limited to citizens due to security clearance requirements: special forces, submarines (philipine island natives allowed), anything having to do with special weapons, etc.
Like Split S said...don't trust the recruiters.