On October 30, 2013, the Air Force changed their rules. It used to be that they would allow you to omit the phrase “So help me God” at the end of their enlistment oath. But no more.
So for now, you have to swear to God in order to be in the Air Force. And indeed, an airman who crossed out the “So help me God” part has been told that if he does not sign the full oath and also recite it (including the God part), he will be kicked out.
Considering that the constitution prohibits the establishment of any religion, and further prohibits requiring religious tests to hold an office or public trust, this will be an interesting court case.
Personally, I think that putting religious beliefs ahead of the constitution is a supremely bad idea. The Hobby Lobby ruling was bad enough, but at least that was not an actual arm of the government requiring someone to swear an oath to God.