My cousin wants to put together a "Mil Spec" AR style rifle. I'm not sure if he wants a 16" or 20' barrel. If you were buying a complete AR Upper Receiver, which brand would you go with? What about the best twist rate for a barrel that will be shooting predominantly 55 and 62 grain bullets?

PSA premium uppers use FN barrels.
16" mid length gas. https://palmettostatearmory.co...embly-no-bcg-ch.html

20" A2.
https://palmettostatearmory.co...ch-carry-handle.html

Kinda tempted to get the 20" upper myself.

AR products come in an unbelievable selection of price and quality. If I wanted a "mil spec" one I'd just get an FN one as that is probably as close to mil spec as one can get. both 16 and 20 are available. But mil spec is 1:7 twist and that would be fine for the ammo weights you mentioned.
Me personally I'd wonder a bit off the "mil spec" range and get a better upper than those (not quality wise but with things like a free float rail, etc.) , but that's just me. I'm not a PSA fan FWIW>

FN or Colt.