Title says it all. Rent due on 10th asked them to leave in 30 days on or around the 8th , now they are leaving "in a week or so" after the 10th. Can they stay without paying rent. What if they leave without cleaning?

Can they, guess they are ?

If they do leave you can sue them in small claims court for the rent.

My opinion is that you got the lesser of two evils. Be glad they are leaving, period. It's much worse when they refuse, stop paying, and wait 2 months or more to be evicted, creating a mess or damage while they wait.
You don't say what state, what the terms of the lease are, who is on the lease, and what the landlord says, or if you are the landlord/roommate. Getting them out under the law might be much harder than 'asking them to leave.'

We live in NH. There is no written lease. It was up a year ago and we've just kept the payments going.

I'm with Joy. Their shuffling out early might be a blessing. How much money are you going to lose if they don't pay rent?