Mutants are individuals with various unusual powers. These abilities manifest at different points in life, some at puberty, others well into middle age. Some may be benign, such as breathing underwater or being able to heal people. Others however, are dangerous. Growing adamantium claws, shooting optic blasts from the eyes, and being able to read minds present problems for mankind due to them being overpowered. For the governed to be able to protect its people, a registration program is needed so they will be aware of what abilities are out there and how to defend against them.

Historically, singling groups out to be labeled for these kinds of reasons has ended badly for those peple. Blacks, Jews, The Japanese in ww2, all ended being stereotyped and abused "for the good of the people". Nevertheless, there are bad mutants out there who will undoubtedly use their abilities for evil and incite chaos in the country. How can the government get mutants to cooperate willingly?

Remember the story of Rudolph the Reindeer

That's not the name he's usually known by though is it? He has an extra descriptor, that being the nature of his being different from the others. The moral of the story of Rudolph is quite simple.

Deviation from the norm will be punished unless it is exploitable.

You've said it yourself, any minority group singled out for special treatment will come to a bad end. Why are you trying to register them? Because little Johnny whose nails grow slightly faster and stronger than normal might be a danger in the future. How is sticking a label on Johnny now going to do anything other than mark him as "other", ruin his employment chances, and drive him towards the end you're trying to avoid?

There is absolutely no way anyone with a basic knowledge of history is going to cooperate with your registration program.

Your solution is to keep everyone ignorant of the outcome every previous such program has had. Keep the population blissfully ignorant and they might just toe the line.

Benefits

In today's economy, all you need to do is providing free health care and free tuition for college/university. Some tax exemptions will go a long way towards that too. Also provide tax exemptions to companies who hire them.

Instead of mutants trying to hide from society, your problem will be that you will have millions of people pretending to be mutants.

"So what is your super power?"

"I can bend my thumb backwards, look!"

"We've been through this before John, impersonating a mutant is federal fraud."

"Come on sarge, it's the only way I'll be able to treat my diabetes and/or get a diploma!"

You don't make it official

Simply put, you don't register them officially, there will be no government mandated registration or anything of the kind, instead you use data sharing to single out potential mutants.

Say, for example, if the mutant's father took one of those DNA tests to discover his ancestry and the latent mutant gene was flagged on the test, the government would be immediately allerted and this person would be put in a secret watching list.

Someone put photos on their Facebook that has gone trough some AI analysis thing-magig and flagged as a potential mutant? Go to the list.

Someone has recently bought a spandex suit and custom ruby-glass glasses on Amazon? Go to the list.

Maybe a mother has googled "I think my daughter is a mutant", now her daughter is going to the list.

Data is being collected all the time from all kinds of sources and this will only increase on the future, the government won't have problem accessing it and using it.

From them on the government can use the excuse of national security to scrutinize these people's entire life just to make sure they are not one of the "bad ones".

If you are on social media you will see that many societies interested in accessing the users' personal data and information do not ask directly for "give us your details and your connections", but rather develop a rather dumb game (i.e. what would you have looked like in the Aztec Empire?) and, with the excuse of sharing the results with the contacts on the social network, ask the user access to those data.

Do the same: create a fictitious game which easily give a rewards, ask the participants to provide some personal data for enrolling, one of them incidentally being "which kind of mutant power do you have?", and let the masses play.