@Anonymous --
You're correct. NH does not have a personal income tax on earned income. You should file a non-resident MA return. On that return you should enter all your MA withholdings, but allocate to MA only the portion of your income that you earned by (physically) working within MA. MA will refund the excess withholding. There is no "other state credit" in this situation.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
