I've played around with this a bit more. I think it is just a moody video driver. Or maybe the video card, itself. I'm running 2 different cards right now. There may be a "race condition" that shows up every so often, although I show no conflicts. I had a normal boot up today (in admin), and while I was playing around with the resolutions, etc, the little square came back. But I had swapped monitor A onto video card B and vice versa. The little white square "goes with" the card, ironically the "new" card. When I was done with my settings and switched to "user", the little square was gone.
I'm no longer taking it personally
(It's not a 'bug'. It's a 'feature'.)