Do minions get critical damage, assuming they met the criteria, i.e. the roll is a 20, and the result would hit the players defence ?
Yes, they do.
But, because of they deal plain damage (no damage roll) their "maximized" damage is equal to their base damage.
However, they could have traits or powers that trigger on a critical hit, or they could receive such properties by a nearby leader monster. For example, the Ankheg BroodlingDDI's mandibles (from Monster Manual 2) work this way:
Tulgar SavageDDI (from Monster Manual 3) has this trait.
Sherrok, Firbolg Moon SeerDDI (Dungeon Magazine 189) has an encounter power that makes the target more susceptible to critical hits.
If Sherrok's attack hits, all his minions have increased chance of scoring a critical hit, and their critical hits deal additional necrotic damage.
Also, some Player Character's features or powers can trigger when a monster critically hit (and they also do when a minion hit with a natural 20).
For example the Psion utility power Telekinetic ScreenDDI (Player's Handbook 3) could be activated if a minion rolls a natural 20 (but not if he rolls a 19):