Ian,
Before condemning the control module I'd take a look at the motor in the mirror. For the normal non-indexing mirror motor there are only three wires;- common, vertical motor, horizontal motor. For the indexing mirror there are two more wires coming from a potentiometer on each of the axes. The potentiometers provide feedback to the module so that the module knows where the mirror is.
In my experience I have found two common problems:
Feedback potentiometers
These fail, not electrically as you might imagine, but on the plastic drive shaft from the motor. Although I have tried to repair these I have never been truly successful. The problem is that if the potentiometer lies to the module about the true position of the motor then the module continues to drive the motor past its mechanical stop. This results in the teeth on the nylon rack and pinion assembly being stripped. This could be the source of your squealing noise.
Rack and Pinion Assembly
As described above, even if the potentiometers are repaired the chances are that the teeth on the rack have been stripped by the pinion or vice versa. I have never managed to find a suitable cure for this other than taking the pinion out of a non-indexing mirror motor and replacing the stripped rack in the faulty one.
Hope that helps. I assume you know how to get into the motor but if not, please come back and I'll describe the process of removing the mirror glass.
Cheers,
noz