I live in Nashua NH (rent) and work at the client location in NH but my employer is located in MA. Can I avoid paying MA taxes as I don't make money for any services in MA.

You do not have any MA income ask you employer not to withhold MA taxes.
Only if you are present in MA to work, the income earned during that period is taxable in MA. Your U.S. Tax Return: Working or Living in Two or More states