My husband told me he wants a divorce and he wants me to leave. He put my name on house deed but not mortgage. Do I have to sign over house or can I leave name on it. I would rather walk away penniless than to sign it over to him. Can the divorce force me to sign it over? We divorced once before and then remarried. But this time he protected me by putting my name on deed so he could not just throw me out like last time.

No, you don't have to sign over the deed to him (unless it's how you settle the divorce). The mortgage has nothing to do with it.
You have divorced before, so you know that a lot of what 'must' happen depends on the terms of the divorce. A few states are community property states, and have special rules about assets that are acquired before and during the marriage. But once your name is on a house deed, you are a half owner.

I realize that emotions are running high, but you aren't as rich as Liz Taylor and Richard Burton. Marriage counseling would be a lot cheaper than 2 marriages and 2 divorces from the same man. And hire someone to turn the house into a duplex, or just divvy it up in such a way that you each have your own space.

With your name on the deed, he can't force you to leave and he can't force you to sign the deed over. However, a divorce court can, once the divorce is final. Its all part of the settlement negotiation.