Aux belt needs adjustment
The compressor may have locked up, do you hear a squeal before the engine shuts down?
bad clutch bearing
it will scratch itself with its foot all the time and will squeal loudly and throb its body
That depends on how, when and where it is squealing. For example, if the brakes squeal when applied, the brake pads are probably worn down. A squeal from the engine compartment whenever the engine is running could be a loose serpentine belt (especially if the squeal changes a little when you rev the engine).
First blush guess: the compressor is siezed, so that the engine can't turn it. If you rev up the engine a bit before you turn it on, you might find that the engine will load, but not die, and the belt will slip/squeal/smoke.
Probably the brake pads. These squeal loudly to warn the driver of impending pad erosion and urgent replacement of the same. If you continue without replacing them the squeal will get louder and eventually the brake disc (or rotor) will get damaged.
I would suspect a worn or loose drive belt. Can also be an alternator or A/C compressor bearing failing.
Yes
shriek, blare, cry, howl, screech, shout, shrill, squawk, squeal, wail, whoop, yell
squeal is a lose belt or one of the accessory will need to be replaced soon
Replace belt self tensioner, if applicable. Adjust belt tension. Replace belt. Have ac pressures checked. AC compressor may be failing.
Likely either the serpentine belt is squealing due to age and the extra stress put on it when the a/c compressor is activated, or the a/c compressor or it's pulley are failing.