Well, this is strange. I'm on my mother in law's computer now, and I've at least found something that can replicate the symptoms every single time. When I booted I noticed that AVG was claiming an error, and it looked like the update service had experienced a problem. I tried running the update and the computer crashed. No bluescreen, just an instant reboot. For troubleshooting purposes, when the computer came back up, I tried that again and the exact same thing happened.

However, I don't think the problem is AVG's fault. That isn't the only thing that has caused the problem, just the only one I've found so far that's repeatable. For example, when I started writing this post I went to check the event viewer, but once I clicked on "manage," the computer rebooted. I was eventually able to get in, and there were zero errors or even warnings in the event viewer. Nothing out of the ordinary.

I'm not sure what AVG is doing when its update service is running, but it must be something the computer doesn't like.

*edit*
Well great. After I posted I tried updating again and it worked fine. I don't have a clue what's going on here...