The A/C compressor adds more of a load to your engine. The engine could possibly need a tuneup. Try changing your spark plugs & wires.
Check the alternator or the main lead of the ac wire to check for either if there is too much pressure is being applicated to the alternator or if theirs a shortage in the main lead wire to the ac
Check the temp sensor. Also, if the AC compressor is on the radiator fan will be on, so turn off the defroster and the AC.
Loose or worn A/C drive belt. Tighten or replace. If serpentine belt, replace.
I would search the local auto wrecking yards for a 2004 Neon that has factory air then strip all the necessary parts from the donor car to make yours an ac car.
power steering fan belt. It happened to my neon whenever the air was on. Try driving without the air on, if it runs fine then it is the fan belt. Mine squeeled when i turned on the ac and when i turned...$10 for a new AC belt and $40 service fee to have it installed. My car no longer sqeeks, and the AC is WONDERFUL!
It uses R134a.
R12-if it has ac of course.
See if you are getting voltage at the relay when you turn the AC on. Check your wiring and connectors before replacing PCM, I doubt that is your problem.
The ac pump may be seized up.
It doesn't have a cabin filter.
The Neon is charged with 134a refrigerant.The amount used and type of oil are important. Take it to a mechanic.
High Voltage AC.