Do you overclock any components? Is there sufficient airflow reaching your computer parts? This might be the case of your CPU overheating and your computer shutting down to stop damage. Same reason for your wonderful GFX card; would hate to see that overheat. Seeing as though your manufacturer is Gateway, the case the components are in is probably not well equipped to properly cool either of these components. Especially if your mobo is passively cooled with its heatsink.

FFXIV demands high performance and that means high temps on those components. If they are not cooled properly, they will be damaged. Look into getting a case designed to cool your components properly, and get a dedicated heatsink for your CPU.