I don't think a single random shutdown, without additional diagnostic info or indication of other problems, can be considered a problem requiring service. Especially with games, there's a lot going on and all kinds of software conflicts can potentially wreak havoc.
The power jack issue seems more problematic to me. I had an HP with a power jack like that, and it only got worse, to the point that I had to tape it a certain way to get it to make proper contact. The jack itself was coming loose from the M/B.
Unfortunately, I think both the "refurbished" and "Best Buy" parts of your description do limit your Asus warranty coverage. (Even new, BB models are tied to BB coverage, and some of the Asus warranty features don't apply.) If I were you, I'd go to Best Buy and see if they will fix the power jack or replace the computer.