Can my ex legally sign the house over to me even though my name is not on the mortgage?
Yes, Your spouse can quick deed the house over to you.
Your EX may have you re-finance to make sure he/she has no financial responsibility once they sign over to you.
Title and debt on real property is different. Yes, your spouse can quitclaim the property to you, but your spouse will still be responsible for the mortgage unless the property is refinanced in your name as well. Deeding the property without paying off the mortgage (either through sale or refinance) may be a breach of the promissory note your spouse has with the bank which could cause foreclosure issues. A quitclaim deed alone will not protect you. If you do not have a settlement agreement or a court order dividing the property, your spouse could still ask for 1/2 the equity in the residence through equitable distribution even though he/she quitclaimed the property to you.