HELP: DVD player issues

Okay, so last night our DVD player suddenly starts displaying a big black square in the middle of the screen whenever it tries to play a DVD. It's only a cheapie but it should still actually function, right? We've only had it for a few weeks and it was fine up until now. (The previous one, a diferent model which lasted a couple years, gradually lost the ability to recognize when there was a disc in its tray, so we replaced it with this cheapie currently in question.)

Anyhoo, this mornin I took the DVD player back and managed to get it replaced with a brand new copy of the same model. Brought it home, hooked it up: right fresh outa the box, this one is displaying a big black square in the middle of the screen whenever it tries to play a DVD.

Bear in mind the following points:

- when I play an audio CD in this machine, the screen has a track listing and whatnot but there is no black square in the middle.

- when I go upstairs and hook the DVD up to Hunter's TV, it plays DVDs with no black square in the middle.

- our main downstairs TV displays cable, and VHS tapes in the VCR with no black square in the middle.

So it's not the DVD player, it's not the TV, it's just the combination of those two. It's just like boom, suddenly this TV decides it doesn't like DVD. Yesterday morning the old DVD player was fine, but last night it suddenly starts with the black square business, and its brand new replacement is doing the same thing, but just on this one TV.

Does anyone have any clue what the eff is going on here? And (ideally) how to fix it, please?