Why is my Dremel is making loud squealing noises when in use?

The bearings are dry and/or have run out.

I'm not sure about this model, but historically Dremel tools use bronze bearings. These are copper-colored rings that hold the shaft in place at both ends. They are slightly porous so they soak up oil. Over time they become dry.

Split the case to reveal the bearings. If they are ball bearings, they must be replaced. If they are bronze bearings as described above, and they are not worn because they've been left to run dry for too long, these can be lubricated with light machine oil. Use a fine nozzle to put a drop of oil where the shaft meets the bearing. Spin the shaft by hand, you'll see the bearing "sucking" the oil in. Repeat until the oil is no longer absorbed into the bearing, and clean excess with a cotton swab.

Reassemble your Dremel and let it run on low speed for a minute. If it runs silently, gradually turn up the speed. If all is well, it should run smoothly. If it still squeals, the bearings have been damaged and must be replaced.