My cousin did not sign a lease. We had a verbal agreement that he could stay free for 2 years for cleaning the place up after a tenant had wrecked the place. That was 10 years ago and he won't move out. Can I evict him? And how?

You may start charging him rent if you want, or ask him to move.

Merely give him a notice to move and if they don't move file eviction

He is a resident, which conveys preety much the same rights AND responsibilities as a tenant. This means he HAS to be evicted the same as any tenant.

Check with your local housing court for details on how.