No, you are not required to file a New Hampshire state return. The state of New Hampshire does not have an income tax. New Hampshire residents only file a tax return if their interest and dividend income exceeded $2,400 ($4800 for joint filers) plus additional exemptions for age, blindness, and disability.
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The general rule is: your report all your income on your home state return, even the income earned out of state. You file a non-resident state return for the state you worked in and pay tax to that state. Your home state will give you a credit, or partial credit, for what you paid the non-resident state. So, there will be little or no double taxation. Do the nonresident state return first.
Since NH does not have an income tax, you do not have a non-resident return to file. But you still have to pay tax on that income to your home state (ME). If your NH employer did not withhold ME tax, you will owe it, at tax filing time.
If you're a Maine resident, and your employer is not withholding ME taxes from your pay, you should make quarterly estimated tax payments to ME to avoid any penalties.