Thanks for all the hints and tips on here. Like other's, I've had issues on my Zenbook UX21E for years, while it was running Windows 7. I never upgraded the OS for years, other than the traditional MS updates. Under windows, the issue was confined to sudden shutdowns while pulling the AC power. (Honestly, I just thought it was a short in the power connector, so I would just shut the unit down before unplugging.)
However, last week I put Ubuntu 14.04 on this machine, and the behavior got worse. Not only would it shut down when the power was pulled, but also after about 20 minutes of use. I installed sensors and stuck it in a while-do loop, and forced the machine to work hard by running Fraqtive animations.
Sure enough, when the power was less than 75% AND the heat was 80C, the machine turned off.
After reading this thread, I updated the firmware - that helped with the sudden shutdown after pulling the AC, but not when the 75%/80C limit was hit while on battery. The machine would still unexpectedly power off.
So - I opened up the back, and disconnected the battery for 5 minutes. After reconnecting and powering back up - I ran the sensors and Fraqtive test...
....the machine stayed up until the battery ran dry, regardless of the CPU temperature.
So, best guess is that it was a combination of issues: the outdated firmware was responsible for the sudden shutdown when yanking out the AC (for all the reasons stated above), and the disconnect/reconnect of the battery clearly reset some sort of low battery reading that was inaccurate. (Whether it was the firmware in the battery or the firmware in the Zenbook that was issuing the false positive, I couldn't say.)
At any rate, I'm back, and the machine is quite healthy. So, thanks again to everyone.
Rob