Mine has done that after the power has gone out, too. It indicates that you've had a general hardware failure. It's kind of an "oh shit, don't know what happened, but it's not good" from the 360. After the RROD, mine has booted right back up again after I turned it off.
I'd say don't mess with it. Keep it as it is. I've heard that it is a very bad idea to hook your 360 up to a UPS--that would be the only way to fix your problem. In fact MS 2nd tier support as well as people in the forums told me not to hook up my 360 to a UPS. Supposedly that huge 5 lb. power brick that you plug in has surge protection sufficient for the 360's needs.