Actually, once the British pulled out of Boston (17 March 1776), that was the end of battles fought in Massachusetts. In fact, there were only two battles of any consequence, both very early in the war: the fight on Lexington Green and the subsequent retreat of the British to Boston, and Bunker Hill. There was some action around Providence, Rhode Island later in the war.

Massachusetts was one of the most ardent supporters of the Revolution among the colonies, and her soldiers did in fact appear in most of the campaigns of the war. But refugees? I don't think so. More likely people fled TO Massachusetts. The real northern battleground colonies were New York, New Jersey, and Pennsylvania.