What can you spend these on when rolling your Discipline check to use certain aspects of Move and Influence?

Can you score a critical with Move?

I would say you can really use most anything that you would with a typical weapon. Advantage for boosts to players, recovering strain, Triumph to have whatever beneficial thing you can think of (maybe one of the objects thrown bounces off the enemy and hits a nearby enemy or even explodes on impact). Threat could be that the player took strain because of the intense focus, or maybe even they grabbed the wrong object on accident, and it was really whatever box a nearby ally was hiding behind. Despair is really the tricky one since unlike with ranged weapons, you can't run out of ammo or slightly damage your weapon. It's probably going to be something a lot more dependent on what objects the person throws, but a generic one could be that the attack causes the enemy to instinctively fire a shot and hit an ally, or that the object hits both the enemy (on a success) and an engaged ally.

I would say you can score a critical with Move, and use the improvised weapons critical rating (which I think is just 5 advantage/ a triumph).

That's for Move, but what about for Influence?

I guess my question is really, is there anything the special rolls can be used for that's central to the use of the Force power, or is it just the sort of generic stuff you can spend it on?

If you're using Influence just to try and boost social rolls (left side's control), then you'd just do anything you'd do otherwise. For the right side control, it probably depends on what they're trying to do. So let's just stick to the example of them tricking a guard into letting them pass an Imperial checkpoint for this. Advantage could mean that he let's the players pass through and give them a heads up for any other checkpoints in the area - or can be used to recover strain to represent how the character is relaxed from passing the checkpoint. Triumph could mean he calls into his comms to let the players pass through all the checkpoints, or even thinks the players are high ranking officials and lets them browse through a locker of confiscated weapons. Threat could mean he still let's them through but won't allow any rifles - or alternatively strain is taken to represent the focus put in to trick the guard. Despair could mean he takes all their weapons or alerts other checkpoints to watch out for the group.