St Croix Island
No, not in the Virgin Islands. St. Croix Island is a mile or so off shore from Calais, ME in the St. Croix River (tidal river). The river marks the boundary of the US-Canada in this area of Downeast Maine. The historical site is maintained by the National Parks service but, it is territory of both the US and Canada. NPS has a historical information center and boat launch in Calais. It is from there that I have kayaked many times to the island. There is cobble beach to land on one side of the island (rocky edges otherwise). Above the beach there are meadows and a few trails. LNT is heavily practiced here so, visits are allowed but, no camping, no facilities, do not leave trash, and do not take anything from the island.
I do not really know the history of the island but, I do know that there were a few archaeological digs there in the past. Here is a link to the webpage if interested. https://www.nps.gov/sacr/index.htm