So just an update on my side. After having my computer restart four times yesterday I got to talking with someone in-game about it, which reminded me that I had updated my Nvidia video card drivers to the newest beta because someone had mentioned it was giving better performance in XIV. I decided to rollback my drivers last night, and I haven't (*crosses fingers*) had any restarts/crashes today.
So, assuming my computer doesn't restart right after I post this, I think the problem (at least in my case) was either a bad installation of the drivers, or the newest Nvidia beta driver does not like FF14.
"Your fate is my destiny. And your destiny shall be my fate."