Considering that one of the insufferables was Great Britain's agreement with Native Americans/First Nations regarding the Ohio region, Native Americans had far more to gain or lose in the war than most if not all Americans themselves, and many of them were aware of this. The United Empire Loyalists who emigrated to Canada following the war were made up of primarily religious minorities who feared the new government (who did not appear very dedicated to religious freedom, especially for Catholics) and First Nations (at least they're First Nations today, since they're in Canada!). In particular, the Mohawk fought under Joseph Brant, who also led much of the other New York loyalists fighting, and after the war they left the country and moved to Six Nations in Canada.

That said - Six Nations in general supported neutrality in the war, and many groups wanted to side with the revolutionaries, under the mistaken belief that freedom for one was freedom for all. The peace that followed entirely ignored the sovereignty of the Native nations of North America, and enemies and allies alike were put on reservations and had to give up most of their land. this also led directly to what happened in the war of 1812, where every single Native nation bordering with the Americans between Canada and Mexico joined up on the side of the British, only to again be ignored when peace was negotiated.