The reason I asked if it stopped when downward pressure was put on the pick-up, if it is a backing plate or loose mounting hardware vibrating, then that would kill the vibration and it would be a easy fix.
The reason only one p/u is squealing is because only one is bad.
So best case, loose hardware, easy fix, 2nd best, loose winding, p/u needs to be potted, worst case, potting sometimes doesn't work, p/u is bad and needs replaced, it is a 35 year old p/u.
Disclaimer, these are my best guesses, it is very hard to get it right over the internet everytime, good luck and good hunting...........