So...during my troubleshooting, I took the Quad switch apart. I found that the input jack has a small PCB on the back of it, and that it had turned a bit the last time I tightened up the nut on the jack. It had turned enough that the small PCB on the back of the jack was touching the large PCB that controls the Quad switch. I can't confirm that it was making any kind of connection, but I put a piece of electrical tape between the two to be safe and tightened up the jack while holding the tiny PCB straight.
When I put the box back together, the Quad switch works fine in my original GX-700. It still is not working in the one I just picked up recently. I'm assuming the second GX-700 is broken, but I want to try a few more things before I give up.
My midi channels are set the same, my id numbers are the same, etc. etc. Each patch has the correct CC commands to turn individual FX on/off, but for some reason, they're not receiving the message from the switch. There has to be a setting in there somewhere that I'm not finding (yet)