OK, I fixed it.
When I was down at my grandma's house helping her with her computer, a message came up on MacAfee alerting about a new virus, W32.IRCbot, which causes random reboots. So I went home and did a search (from my computer) for that file, and found nothing. Then my new case came (about time....stupid UPS) and I installed it into the case (BTW, see my other thread about that) and then booted it up into regular mode, just to see what would happen. Avast immediately came up with a message, saying that the virus W32.IRCbot was found, and I moved it to the chest, and the computer has been running since.
Yay! Now if I can only fix my power supply....