Never, and I mean never, remove a cable from the battery or alternator while the car is running. Good way to burn a diode. If you're done this, then take the alternator to Autozone or whatever and have it tested. Maybe you're lucky.

The reason the belt squeaks only when the battery is charging is that the alternator is pushing a lot of current through the battery. High load results in more back EMF, which is a force which opposes the rotation of the alternator, so the belt system has to work harder. Either the battery has a shorted cell, there's an undersized pulley, the alternator has a bad diode, or it's oversized. It's also possible (but not likely) that there's a short elsewhere, so do 911's glow plug test. You said the alternator was new, but was it stock sized?