If the house is in my husband's name and we divorce, would it go to him?

If the house was purchased during the marriage, then the house is "marital property".  Both of you have an equal right to the house.  In a divorce action, there are basically three options: (1) the husband gets the house and pays the wife 50% of the equity to buy her out; (2) the wife gets the house and pays the husband 50% of the equity to buy him out; (3) the house is put on the market and sold.  The parties then share equally in any equity or debt associated with the house.  There are other options depending on the assets and debts of the parties.  I would need more detail in order to advise further.