The reason you (can) have 20 in your squad is for when people die.
If one of you crew goes down, then you only have a couple of turns to get to them before they die. If you get someone to them, they call the medic, who will steam in and carry them off. This will ensure they get saved. If you don't do this and they die, then that is it and you won't be able to use that person again in that particular game save (as they are dead).
So say you have 20 in your squad, but 2 get killed, you can then go to HQ (assuming you win the battle) afterwards and top up your team with new recruits. The list of new recruits increases as you progress through the game, and this is where I expect Vyse will show up at some point.
You don't actually have to have 20 in your squad if you don't want, but I don't really see any reason not to.
Also, some of the soldier types are fewer in number than others. Snipers seem to be particularly thin on the ground, so you want to be pretty careful not to let your snipers die, or you could find there is nobody to replace them.
PSN - hanafuda
