It does sound like a dead, hot or stuck pixel problem. All cameras will do this to some extent particularly with long exposures. If you shoot in RAW mode, then DPP or other programs like Adobe Photoshop, Elements and Lightroom will map them out automatically. Otherwise, you can contact Canon about having a pixel mapping done where they map out the bad pixels. I have heard of them even doing this for free out of warranty (minus shipping) but don't count on it. Either way, a trip to Canon for your camera might reveal a much cheaper fix than a sensor replacement.
There is a method for doing your own pixel mapping that some (myself included) swear by and others say is BS. I have used it on several cameras and I am convinced it works. You simply put the camera into manual cleaning mode with the lens cap or body cap on (you don't actually go in and do a cleaning) for at least 30 seconds. Do a before and after of a dark (cap on manual exposure) to see if it is working. My 7D had two red pixels on long exposures right out of the box. I ran this mapping two or three times and they are now gone. Good luck!