Bad sqeel or screech w/defrost mode, when compresser is engaged. turned it to just heat it stoped

How did you test the A/C system ? Hook up gauge manifold , test A/C pressures ? Manual or automatic climate control ? The blower motor only working on high means the blower motor resistor is bad ! Did you hook up a scan tool an check high an low pressure switchs inputs to the PCM - engine computer . Does the A/C compressor clutch engage ? turning on the compressor ? Moving the mode switch does nothing . Doesn't move from floor to vent to defrost ? Probably the mode door actuator ! Don't think the HVAC control panel is bad . Just a few little things . Pushing the A/C button on the control panel just send a electrical signal to the PCM , the PCM will energize the A/C clutch relay , turning on the compressor .

The Trainer 40 When the compressor won compress

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The heat/vent blend door that is located inside of the HVAC system behind dash..Rarely it is the control however it could be....Some of the vent door issues are recalls on certain vehicles."Also Blend vent door actuators that operate the doors can become inoperable...If it is the door inside the HVAC it will probably take about 8 hours to repair...Most vent doors either stick shut and or break..as well as actuators....Only a Dealer Scanner can tell exactly where the problem is.....

Depends.
The compressor will run when the control is in defrost, but should not in heat mode. In defrost mode the compressor runs to dry the air blowing onto the windshield.

If by heat you mean defrost, it could be a seized AC compressor or compressor clutch, since the AC compressor is also engaged when in defrost mode to remove moisture from the air.

Try turning the flat, round plate on the front of the AC compressor by hand (WITH THE ENGINE OFF!!) If turns with only a little resistance, the compressor is most likely NOT your problem. If it not turnable by hand, then your compressor is seized up will most likely need replaced.

Your secondary cooling fan motor is bad and needs to be replaced. When you turn on defroster or a/c this motor is commanded on by the computer to keep the engine cool under the extra load.

The A/C compressor is supposed to run in defrost mode. The only modes the compressor won't cycle is floor only and face only. This is to keep the seals in the system lubed during the winter.

If you don't have heat, check the coolant level. If the level is low you won't have heat.

a/c compressor may be seized. A/c compressor kicks on when in defrost, a/c, floor/defrost modes any time temperature is above 40 degrees F. That will burn the sepenine belt and stall the engine.
Try turing heat on floor mode only & see if ok.

the sqeeling suggests that the drive belt/s are slack and req readjusting however considering age of car may i suggest that you renew belt/s especially because the belts slipping would have caused damage plus unnecessary wear due to o/heating by slipping i'm sure you would not like to break down due broken belt ??

See if it goes away when you turn the A/C off. The a/c comes on when you sre in the defrost so yo don't hit the windshield with hot air. The screaching is more than likely the A/C clutch trying to engage, that why it was overlooked or ignored by the mechanic. It is a common thing. The battery should be allowed to recoup from the starting of the car which a few minutes of running woll help so the a/c clutch gets full battery voltage to function.. Turning on the defrost and /or the A/c after about 3 minutes of warm up. Open your hood and have someone turn the a/c on and off, you can hear and sctually see the a/c clutch doing its irritating thing.But keep your fingers out off the engine and don't lean over it with loose clothing. Good Luck, Ned

Both your air con and defogger (I am assuming you mean front windshield defroster setting, and not the back window) engage the air conditioner pump/compressor.

The screeching can be either a bad belt or a bad compressor.

How long has it been since the belt was changed? If it is old, and slipping, it will cause the screech at speed.
If the belt has recently been changed, and is adjusted properly, I would start looking that the air compressor pump.