I wanted to get a roommate with the approval of the landlord. I got the approval and accepted a deposit. At the time they were going to use the deposit from the last place to pay rent. I allowed them to move some stuff into the house so that they could close the deal on their old place and get the deposit. It was late at night when they moved some things here and I allowed them to stay the night. They have refused to povide rent and refused to sign a lease and now do not want to leave claiming they have a legal right to stay here since I allowed them to spend several nights here. Is this true? I thought I was allowing them to stay as a guest and they have worn out their welcome when they have not made preparations to stay legally. I have offered them full desposit to leave but they still refuse.

Location is anchorage ak.

By accepting the deposit and allowing them move in, you established them as tenants, not guests. There fore you will need to formally evict them. Since they paid a deposit there is a clear intent to rent so you serve them a 7 day pay or quit notice. If they don't leave within 7 days, you have to go to court for an eviction order.

Yep, the moment you made the deal and accepted money they were tenants. You must now start the steps for eviction following the laws which apply to your state. Where do you live?

Each State is Different. Check this link out http://dced.state.ak.us/occ/pub/landlord.pdf

According to the act A seven day written notice is required to terminate a tenancy whan a tenant is behind on rent. The notice must state the correct amount of rent to be paid. If the rent is paid before the 7 says are up he can stay, you can refuse payment after the 7 days. If you accept partial payment you may have to start the process all over again.
If you tenant pays up then the rent problem is solved, after you could probably give him 30 days notice to vacte because he refuses to sign the Lease.

And if he stays after that then you can start the actual proceeding for eviction. A 7 day or 30 day notice can be ignored but eviction carries the law with it.