Can a character other then Winter Soldier occupy the Sniper Nests? I didn't see any wording saying otherwise. I'm just looking to see if I can moves characters into the empty nests and prevent Soldier from moving into them if I move another character to go after Winter Soldier.

I believe it would work that way, which seems kind of thematic. Sending agents to watch all of his hiding spots, flushing him out into the open. Very Cold-War-ish.

Exactly, even if can't, if you just stand next to the nest, you can have teams waiting for him at the other nests just to switch.

Sure can.

Oddly, occupying the square wouldn't itself trigger the "discovery" roll(, even though moving adjacent to it to move into it would).

Oh, but if you occupy the nest and he is required to be placed in that nest, he will have to be placed adjacent to his own nest, won't he?

Agreed.
Why do you say that?

Because the nest square is occupied and he is required to be placed in it and the golden rule of occupancy. Was I wrong in my conclusion?

I'd say it should be that he doesn't move at all.

But I'd also say it requires clarification.

If he is knocked off his markers in some way, but unoccupied markers remain on the map, can he still be placed in one of them while he doesn't occupy them "once per turn"? I don't see any reason why not, but I thought I'd ask because someone is going to.

I don't know. My first thought is that he wouldn't be placed anywhere and/or he would be Placed right back where he is as Golden Rule > Special Rule.

For Thanos (Chase) I know it was first ruled you could place the figures as close to Thanos as possible, if being placed Adjacent wasn't possible. Now the PG states you cannot use the power if they can't be placed adjacent. Are there other precedents? I can't think of any but thought maybe you could and that's why you felt that way.

But Chase Thanos did let you place as close as possible until the PG clarification changing it, so that would lead me to believe you just place as close as possible. The other two precedents I can think of are the carry ability and Bat-Mite. Bat-Mite is still in the cave and the carry ability precedent is simply "just don't try it".

Oh, and the Golden Rule for reference:

Yeah, clarification would be good. Forcing the roll with each use of LC or Sidestep would be damned annoying.

This trait is really cool but with little loops

Like if you send 3 cheap pieces or even bystander tokens each to one nest then Winter soldier is neutralized and can't do anything anymore.
He needs to be sharpshooter or maybe another power action to leave the "nest"

This made me think of (updated) Spiral with an obstructed gateway.
(Insert your own joke here. Heh, "insert.")
(Not that they are perfectly identical, but discussion of both involved The Rule of Occupancy.)

I find it funny I was the one to answer that and then essentially asked that question I answered here.