My husband was in a a auto accident in MA caused by an uninsured NH resident. How does he receive compensation for damages for his vehicle, etc.

The first thing he should do is seek all necessary and appropriate medical treatment.  Then he should immediately notify his insurance carrier.  He should specifically notify them that he will be making a claim under his own policy's Under/Uninsured Motorist Protection Provision, if any.  You may want to get a lawyer.