Hi C.Saenz, Turn the A/C on, and immediately put the clutch to the floor, then release, then back down, release, back down for a few times. See if the noise changes.
Like your saying it happens when shifting gears, I am leaning towards a worn clutch release bearing. These squeal and also grubble when worn, but mainly squeal. When actuating the clutch the noise should get louder.
This part is relatively cheap, however while you are in that area, you might as well buy a clutch kit - so clutch disc, pressure plate and release bearing.