I don't want to alarm you but the same thing happened to my aunt. It all started with a boil under her right eye which caused her severe headache and pain. Pus came out the next day but another boil grew near the first one. When she consulted a doctor she was informed that she had several boils beneath her right eye which may cause her to lose the eye when they finally burst. And it did happen. So I suggest that you go to the doctor immediately to get the proper treatment before something serious happens.
It doesn't sound like a normal zit.
Squeezing could be spreading the infection. If it is an abscess the spot will fester until the core root has been removed and you may require a nurse or doctor may need to remove it.
As the hot compress is helping to bring the rubbish to the surface, do this rather than squeeze.
I would make an appointment to see the doctor as you may require an antibiotic to kill off any infection.
A community acquired MRSA type of staph infection-not as common-
is resistant to most antibiotics, but not all.
This is due to the overuse of antibiotics that helps such strains
of bacteria evolve fast and develop abx resistance.
And again it may resolve on its own, after the doctor drains it and cleans it,
(preferred over self-draining)and without the use of abx.
It's the attending doctor's call.
I have already answered this question, on a different post in this community.