So here's the story. At completely random times during my game with seemingly no pattern to it - and only during Cortex Command, no other application on my computer does this, not even more CPU intensive ones - it will suddenly switch off and my entire computer will restart.

This happens with no other program on my computer, and as I said, it seems to have no pattern whatsoever. Sometimes it doesn't happen at all after hours of play, but the closest thing to a pattern I can find is that it often happens roughly 15 - 20 minutes in. (Though it does happen at other times as well, just not as often.)

I only have one 'mod' and it's a single custom soldier of mine using a heavy coalition base template.

Anybody have an idea as to what is going on?

What applications do you have that are more CPU intensive than CC?
I've never heard of this happening..

This computer is designed to play games from this generation on maximum settings with very minimal loss in performance. When I play Cortex Command, it doesn't take up any more of my CPU than any other game I play.

Most other games use your GPU much more intensely than your CPU. Especially games from this generation directX 10 and whatnot. CC requires directx 7 or 8 or something, and does everything on your CPU. The only thing I'd see as comparable would be compiling a huge project or doing any maths based drawing (FLAM3 and the like).
That said, it shouldn't straight up shut down your computer unless you've got a piss weak CPU and cooling system.

Especially since it didn't start doing this until the most recent version of CC.

Could it be a power supply problem? If you have a serious cooling system or an inefficient CPU you can over load the PSU and this sometimes causes reboots.
Keep in mind CC is not optimized, uses no GPU at all, and is very intensive heat-wise. If you upgrade the computer but use a lower wattage PSU this can happen. I only know this from overloading PSUs this week. Ended up with two of them, to make it to 500W.

If that's not it, use a stat recorder to see if anything anomalous happens right before the reboot.

Your heatsink, my good sire, is faulty.

Same thing happened with one of my family computers. It would work fine until something was started, then it would shut down. I busted open the case, and what do you know, the heatsink was broken off! Mystery solved.

Also, what are your specs?