AFTER.

Get through UPT. Age limit for UPT; no age limit for a Master's. If you enlisted (non-officer) with a Bachelor's degree, just to get the GI Bill, any officer board would probably wonder "Why?"

Unless you truly want to work in the engineering field, the subject matter doesn't matter (pun intentional). Most guys who go to pilot training aren't really interested in doing short-term pilot, and longer-term engineer.

I was an idealist too. My degree is in Engineering; I wanted a Master's in it too. While it would have made study more relevant and interesting for me, it wouldn't have helped my career. So, I took the closest thing I could get through the (rampantly available and somewhat dubious quality) Master's Program available at every military installation in America. It's a scam for promotion, and everyone knows it. On the other hand, I think some real brick-and-mortar schools have legitimate extension Master's Programs, if you want to get the real McCoy.

Incidentally, I tried to get a job as an engineer when I got furloughed. No-go. Said I didn't have the relevant experience.....and I've done some engineering work for the Air Force (in a non-official capacity).

Studying for a Master's once you are established in the military is tough. But it is of lesser importance than getting your foot in the door.