If you are living in someone's house and not paying rent and they want you to leave, do they have to give you a notice, and also do I have any kind of rights to stay there?

If you have lived there long enough to establish residency (such as receiving mail there) then you must be treated like a tenant. In Florida that means that the landlord must give you 15 days written notice that your tenancy is terminated and you must move out. If you don't move out within that time then the landlord can file a lawsuit to evict you.

No you have no rights to stay there. There is no such thing as squatters rights. All you have is the right to 15 days written notice as Lisa said.