Criticals are technically only automatic when applied to attack rolls.
This is the correct answer, as indicated in the other answers here.
It is not uncommon for a group to respect the Nat 20 and Nat 1 as auto success / fails, or close to it.
At my table, if a player rolls a nat 20, I give them the best possible result I can without breaking the world.
Example: A PC rolled Nat 20 on his Arcana check some on magical water to know what it would do when he drank it, and I gave him the affirmation that it was magical, and of what school, but that's it. Without an identify spell, or actually drinking it, he couldn't possibly know what would happen.
Criticals are great, they're fun for players and DMs, I like to keep them alive through every aspect of the game in as much as they make sense. That said, auto successes / fails do work a lot better with attacks and saving throws, and are a lot tougher with ability checks and contests because of how varied they are, in application and situation.
A good rule of thumb is not to allow rolls at all for the impossible. But say a character gets bit by a poisonous creature with a DC of 21 and has a -1 to his con check, what a great story moment to pull out a 20 at that moment and succeed against all odds! It's not impossible to pass a DC 21 con check. It would be impossible to say, jump a 1000 ft cavern.
Above all - D&D should be fun. Respect the rules, but find the balance that works for your table.