First, when you experienced the squeal, the loss of power steering, and the battery light flickering - at that time the serpentine belt was not doing its job of transmitting power to those items. It was most likely slipping on the crank pulley because the a/c compressor had locked up. You are lucky that you were only a block from home at the time because when I had that happen to me two blocks from home, the belt broke before I made it home. You have seriously damaged your serpentine belt and it needs to be replaced post haste to avoid being stranded somewhere. If you look over the belt carefully, you will find a spot that has been badly overheated.
The waterpump was not turning in the situation above, so you were not circulating coolant during that time. It is unlikely that the motor overheated in one block. Most likely the coolant being low was coincidental to the a/c problem, but it could have boiled some out after you shut off the engine.
If I understand the next part of your story right, you let it all cool down, the car seemed to be normal without the a/c running and also with the a/c running. That suggests that the a/c system is low on oil and the compressor is not getting enough lubricant, or the compressor is just about to die for some other reason. So, I would recommend getting one of those cans of a/c oil and putting that into the system, with hopes that its just low on oil. If that's not it, you will again experience the compressor lockup in the future, and it will continue to get more frequent until the thing doesn't work at all. Be aware that when the compressor seizes up like that, it is putting metal shavings into the a/c system, which is very bad especially for the expansion valve / orifice tube, because that can get plugged very easily from some of that debris. Therefore, the textbook thing to do is to replace the compressor, replace the orifice tube, and clean the condenser and evaporator. However, in my experience you have at least a 50/50 chance of saving it by adding oil now.
You must be ready to turn off the a/c at once when you hear the belt screech or you will ruin another belt. If the belt breaks, you have no waterpump circulating coolant, so the engine will quickly overheat; you will have no power steering, so the car will be a beast to steer; and the battery will go dead because the alternator will not be charging it. You need to avoid that situation.
If you add oil to the system and the compressor locks up again, then you need to replace the compressor & orifice tube, and flush everything else and start all over.
