FFXIV stresses the system in different ways than most other games. It isn't optimized very well and hammers the crap out of everything at once. Not uncommon to see systems become unstable due to voltage dips in a subsystem, or heat spikes, etc.

If you have one of those OC, Special Edition, Turbo type cards---may want to go in to their utility and reduce the overclocks and such. It's a fairly common culprit. Running the card's components faster than the reference design ramps up the power draw and can have strange effects when the system is under high loads elsewhere like with this game. Likewise, if you've overclocked the CPU or Memory you may want to try reducing them to standard speeds and test that angle too. You may also want to disable all unnecessary devices as well--remove extra USB devices, disable the webcam, etc. If it is drawing power from the system it may be contributing to the problem.