check valve just controls the vacuum going to the booster. The booster just makes your pedal "pushable" it assists by, in a simplistic nutshell, amplifying the force you are putting on the pedal to move the piston inside the master cylinder. Without a booster, the pedal would be very hard to push.
I would bet that the master cylinder itself is bad, or going bad and putting the new calipers on is just masking the problem.
The other theory is that the fluid itself is bad. Brake fluid absorbs water so over time, it goes bad. Too much water in the lines would lower the temps needed to boil the fluid. boiling fluid will put pressure on the caliper causing it to seize up. But the caliper would release after letting it cool off for a little bit.

Try this. Drive it til the caliper seizes up. Then let it sit. But leave the engine running so there is still vacuum on the booster. If the caliper lets go in 10 to 15 ish minutes, then you know it was fluid related and all you need to do is flush the whole system out. If it doesn't let go, then you know it is either the caliper itself, or the master cylinder. If all of the calipers were locking up, then I would say it was the booster. It could technically be a combination of the booster and the master. The master is going bad and the booster isn't letting go of all of the pressure keeping a little bit on the piston in the master which, because it is bad, is causing that extra pressure to be focused on one caliper. A bit of a stretch but could happen. Anyway, back to what I was saying, if the caliper lets go, great. If not, then rule out the possibility of it being the master. Take someone's good master/booster from their car and bolt it up to yours real quick. work all of the air out of the lines and go for a ride. If it locks up again, then you know it is the caliper itself (or you didn't get all of the air out, so try to bleed it again just to be sure) You could just have crap luck and all of the calipers are going bad at all about the same time.
Then put your buddies car back together, and get a ride to the parts store to buy what you need and fix your car.