you can and as far as I know once a team does this. claims a player and re-enters onto waivers. I believe if this happens the original team that placed him on waivers gets 1st choice to claim him or not and i believe can afterwards be put on waivers.
No there is no first choice. That is an endless falsehood that keeps getting repassed around. The only advantage Chicago would have is if the player was waived again by the team that originally claimed him off of waivers. If is Chicago put a claim on him once he was waived again. And no one else place a claim except for Chicago. They can assign that player to their minor league team without having to expose him to waivers yet again. There is no first pick because of original ownership. It always goes off of standing for waiver order.
If the team who picked him up was the only claim, then the original team can pick him up and send him to the AHL without waivers, unless I'm mistaken.
Correct. Because the view is, that at that point if Chicago were the only one to put a claim in on that player, he would have passed through waivers to their minor league affiliate with no other claim.