Does everyone on a lease over 18 need to be on it or can it be just one person taking financial resonsibility?

It depends on the landlord and different things. For your own protection it is best for everybody to be on the lease though.
Like if one of the room mates has a party and damage the rug the one on the lease is responsible. Then they have to take the one not on the lease to court and try to get the money out of them.

The landlord said it's California law that everyone over 18 has to be on it. I understand for protection purposes but I want my boyfriend to take full responsibility

I can't find any sources to cite if that's true or not but I've heard that too before. I have to ask, why would you NOT want to be on the lease? It protects you. Say you and your boyfriend move in and he's the only one on the lease and two days later you get in a fight and he kicks you out. You have zero protection because you're not on the lease.