Is it Prime95 24 hour stable, with the current incarnation of its software load? Have you run at least 2 passes of Memtest86 on it?
If you want a quick hardware test, grab
SuperPI Mod and run at least the 4m or 8m tests. But really that'll just tell you if you have a serious CPU or memory issue.
If you haven't tested it thoroughly, then I'd strongly recommend that you grab yourself a copy of
Orthos Stress Prime (which is a dual-core optimized version of Prime95) and make sure it can handle running it for at least 24 hours straight without an error. I'd also make sure it can run at least two passes of
Memtest86 (grab the bootable ISO and burn it to disc and then boot from the disc).
I'm only suspicious of a hardware issue because I haven't seen anything similar at all on any of my systems. If you have really done a complete base-installed version of MC and it's still crashing regularly, while not even playing anything, that leads me to believe that something else is going on on your system. Just because only MC manifests the issue doesn't mean it's necessarily MC's fault. You might just not be running any other software that stresses the system in a similar way!
It's always best to eliminate hardware issues as a possibility first, and then investigate other possibilities.