Q: If I unexpectantly die and my home is in my name only, not my husband's, can he have rights to our property?
A: He will (not "can") have rights to your property before and after your death. You should go to a lawyer and have the deed put in both names.
Q: Will the lender automatically take the home since it is financed under me and he has signed a statement saying he has no rigths?
A: There is nothing automatic in the law. If your husband keeps paying, I doubt that the creditor will care. Just make sure you fix it up where if you go first, that he gets it.