A little grunt lingo there.
They do if they have asked to resign their commission and the colonel says, "You'll be a major if you stay in." Though I was a lowly grunt, my dad was capt., combat engineer '41 duration and reserve to '52. He was offered major if he stayed in. Brother in law was air force major, medical corp., he was offered colonel if he did not offer resignation.
yitbos
You don't have to resign your commission, although you may have to convert it to a reserve commission if you have a regular commission, to leave active duty.
I never resigned mine, even when ARPC (Air Reserve Personnel Center) asked me to. Instead I lost weight, and found an active reserve assignment, which I later lost because I couldn't keep that weight off. I was transferred to the retired reserve list in '92.
Had a get together with some of guys from that last "unit" this past weekend in fact. Not all were guys, but only guys showed up this time ... one of the gals is still an active reservist, and wears two stars on each shoulder. :)
Anyway, the "Captains don't re-up" comment was made because "Re-up" means "reenlist, and officers don't enlist, they are commissioned. They don't serve for a fixed period, although they may have an obligation to serve for a fixed period, after initial commissioning or after a training or education assignment. Right now the AF is allowing separated officers in certain career fields to return to active duty and if they do, they encure such a commitment. (Pretty short one though).