In divorce, can he claim my home and other assets if my husband never paid mortgage, assessment, etc.?

What makes it your home? Was it purchased during the marriage. Was it paid for with marital funds? Title is not relevant. Date of purchase is. It may be marital property even if it is in your name alone. If it is marital property, the fact that he made no mortgage payments may go to the extent of his interest, but he may still have an interest.