This is almost hard to believe at this day and age. Here's why:
When you enlist, you have to recite the oath to the constitution on two different occasions at MEPS. The officer will explicitly tell you that you do not have to recite the part that says "so help me God" if you feel uncomfortable about it. You don't even have to say the word "swear," either, but can opt instead to say the word "affirm."
One of the details on your dog tags is your religion/nonreligion, and no one would bat an eye if you asked to have the word "atheist" engraved on them, because it's actually a default option in the databases along with Catholic, Islamic, Jewish, and others.
I would hope that this is only an isolated incident, but this wouldn't be the first time that the AF has done something whacky since the turn of the 21st century. Shame on them.