[Dr. Dyno]

It may be the belt is just a little loose and squealing while the alternator is recharging the battery for the starting discharge. That's pretty common on many vehicles.
My trick is to spray some water on the belt. If the squeal stops for a few seconds it's loose and not a bad tensioner.

 

[Dr. Dyno]

After the
engine is started & before it gets hot ....
There is a squeal for a second or two after the engine is shut off.

Somewhat confusing for sure! :doh:
I don't know how anything could squeal after the engine stops and it must do that within half a second after shutting the key off unless he means from revved up.

 

[Dr. Dyno]

The squeal happens after the engine is shut off. I start the engine up let it idle for a few seconds (not revving it up) & then shut it off.

Wow, that's a new one on me!

 

[Dr. Dyno]

I suspect a failed Overrunning Alternator Pulley.

Makes sense. It can't think what else it could be. But since the primary reason for the special pulley is during shifts why wouldn't he hear squeal then?