Xbox 360 keeps shutting off mid-game

Hey mate, first things first, check to see if you auto-off is enabled:
support.xbox.com/.../console-turns-off-unexpectedly

If you don't have that enabled, there's a pretty good chance your power supply might be on it's way out. Have you noticed the light on the power source change? Try using the steps here to diagnose the power supply see if it's behaving as it should:

support.xbox.com/.../three-flashinglights

There is also this link:
http://support.xbox.com/xbox-360/system/check-power-supply

With some more recent models of the XBox 360 you can turn off the console mid-download and it will keep minimal functions running to continue while shut off until the download is finished. That is usually what I do with the free games; start the download
before work then turn off the console.

Your xbox itself shuts off mid game or you just lose picture and sound?

Thanks for the quick response, however, I've done what you suggested and even tried a new flash drive and the problem persists. Now I'm thinking it might be a problem with the power supply itself.

More than likely the issue is with the flash drives. PNY is known to have issues playing nice with Xbox consoles. Although I cannot speak for the Lexar flash drive as well I'll bet that may be an issue as well.

I would recommend removing all flash drives and restarting the console. Then play a game and see if the console shuts down. If it does not then turn the console off and add one drive and repeat the process. If the console shuts down then the drive you
have added is the culprit and must be replaced.

I would suggest getting an official Xbox hard drive or an external hard drive (up to 2TB) to store your games and data on.