Did you get it sorted yet? Yes, sounds just like the auxiliary rubber drive belt, the one you can see turning down the side of the engine. Yes, this drives various components including water pump, alternator and air conditioning. If it is worn and / or slightly loose on any car, it can slip on start up, or on gear changes as you have suggested and it will squeal. It is usually worse in cold / damp / wet weather. The belt circuit does have a tensioner to keep belt tight, so belt could be worn and needing replaced if tensioner no longer able to keep belt tight. Worn pulley bearings on water pump, idler, or tensioner etc produce more of a rattle than a squeal. It's quite a durable belt and unlikely to break, but if getting very loose, it may no longer turn alternator and battery may start to lose charge etc.
