OK, I know this may seem like it has an obvious answer but I would like to post this question anyway just in case there is an aspect of Linux that I am not aware of.

For some reason after my server has been running for a day or two, sometimes even longer (up to one week at a time) it randomly shuts down. And it's not a clean shutdown, either. On the next reboot the file system wants to perform an integrity check. I'm thinking this is most likely a power supply issue, however I was wondering if it could have something to do with flushing the RAM periodically, something to do with Red Hat Linux 9, or possibly even some sort of intrusion. I am pretty new to Linux though I am a dedicated learner and have been reading a lot to get things up and running Any input would be appreciated!

its a wiring problem, unless you are running a very fast graphics card and as many bits of hardware you can stuff into a machine with an old power supply.
It also sounds like you are not using a journalling filesystem, I have found ext3 very robust in case of powerouts (I live in an old fishing cottage with pathetic wiring, doubtless due to the difficulty of drilling through 2 foot walls).
in short kick all the plugs systematically and rewire as necessary.

I had an old cyrix 333 that would just suddenly reboot when I used it, sometimes even in the bios, changed m-boards and it worked fine, could be m-board, could be building wiring, could be p/s could be problem with grounding the p/s to the case (I had a p/s that was not touching the case at all and my outlet was not grounded, whenever I turn on speakers or cealing fan, etc it would turn off. (completely off, no reboot)