This sounds similar to the problem that I had with a refurbished 24" iMac. Basically, it would just shut off as though someone had jerked the cord out of the wall. I understand that this is a semi-common issue. The Best Buy folks may even be aware of it, although that is optimistic. Good luck!
I'm sure they'll take it back. Plus it was an open box item so lol. We'll just get a brand spanking new one!