I dont really have any advice other than BLM land and Forest Service land get much less visitors and much less patrols by government officials. In fact I believe you can legally camp for free for up to two weeks on most BLM/FS land, and if you are hidden enough I assume you could stay indefinitely.
Ive got a friend who practically lived on forest service land near in western Oregon but he didnt build any structures there, he just lived out of his tent and his truck, parked on an old unused logging road.
I know another person who is living in the side of a hill in a wildlife refuge and has been seen by workers there but they never threw him out. Again he hasn’t really built any structures either.
I am sure there are people who have gone further than them at making a home on “public” land but they are the only ones I know personally.
Good luck, be careful! If you decide to go for it, tell us about how it works out.
edit- Oh I also know of some anarchists/crusties who squatted an abandoned house on the edge of the NC side of the great dismal swamp (NC side is a state park, VA side is an NWR). As far as I know people lived there for years without being thrown out, they might be there still.
Runaway slaves lived in the swamp during the underground railroad years. They knew they could be killed if they got caught but they did it anyway, we wont be killed for squatting there so I guess in contrast we don’t have any excuse not to do it except that it wont be easy.