Unsuspected shut down?

If heat is the reason for the shut downs, look at the video card first. If the power supply was the problem, the system would be shutting off when running other tasks that tax the hardware more heavily as well, not just when the OP is playing back video.

Keep in mind that when watching movies/videos, after a stipulated period of time, your computer can go to sleep/hibernate/shut down (since it deems itself as idle), depending on your power settings. So do check that.