Some might say "no" to this because mutants don't choose to be mutants. This is irrelevant. If you were born with a gun in your hand that doesn't make you any less dangerous than someone who chose to put a gun in their hand. In order to apprehend mutant criminals, to counsel youth before they abuse their powers, and to maintain an air of safety - it is vital that mutants and meta-humans be registered.
First, guns aren't required to be registered.
Second, it wouldn't work. It would just create animosity and lead to violence.
Listing people categorically is not something a nation can do without opening a few doors which are not easy to close.
The Marvel universe explored this idea during their "civil war" story-line. It was not a fun story-line to follow, as it pitted some of the best heroes against one another.
If I had powers, I would register them, but only under a pseudonym, as a superhero.
My power would be telekinesis.
The approach of labeling a group of people who possess superhuman abilities as "dangerous" and trying to control them, whether by compulsory registration or forcible imprisonment and/or experimentation is problematic. We can't generalize based on fear and suspicision, we need evidence. It's the same as labeling a person who is a computer genius as "dangerous" because he/she could steal money electronically. Just because he/she can doesn't mean they will.
Marking a specific group of people as "dangerous" and therefore needing control over them before they have committed any specific crime is a dangerous road for a country to take. It all starts with identification, then restrictions (curfews), then isolations and consolidation (Japanese in WWII), then of course holocaust. Identity politics leads to animosity and otherization, which causes more violence then it prevents. When mutants don't feel like they're part of society, they are more likely to commit actions that may harm it, just like any other people group.