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:
Mandible Rip (standard, at-will) * Acid
+4 vs AC; 6 damage. If the ankheg broodling makes a critical hit against a target, each creature adjacent to that target takes 3 acid damage.
Tulgar SavageDDI (from Monster Manual 3) has this trait.
The savage can score a critical hit on a roll of 19–20. When the savage scores a critical hit with greataxe, one ally within 2 squares of it can make a melee basic attack as a free action.
Sherrok, Firbolg Moon SeerDDI (Dungeon Magazine 189) has an encounter power that makes the target more susceptible to critical hits.
Ban of the Raven (necrotic) * Encounter
Attack: Ranged 10 (one creature); +17 vs. Fortitude
Hit: 3d10 + 12 necrotic damage. In addition, attack rolls against the target can score critical hits on rolls of 18-20, and the target takes 10 extra necrotic damage from a critical hit (save ends both).
Aftereffect: Attack rolls against the target can score critical hits on rolls of 19-20 (save ends).
Miss: Half damage, and until the end of the target’s next turn, attack rolls against it can score critical hits on rolls of 19-20.
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):
Telekinetic Screen - Psion Utility 6
You mentally project a thin layer of force around yourself to ward off a deadly attack.
Encounter * Psionic
Immediate Interrupt - Personal
Trigger: An enemy scores a critical hit against you
Effect: The attack’s damage is reduced by 10 + your Wisdom modifier.