I once stayed at the mental hospital that I am currently working at and none of my coworkers know.
Do you think they would judge you if they knew you stayed there before?
Yes. I want to be normal to them. What kind of crazy person has to stay at a mental hospital?
I don't think there is anything wrong with staying in a mental hospital. It is a good thing to receive mental health treatment when you need it. You shouldn't be ashamed of the time you spent there.
I still am too embarrassed to tell my coworkers about it.
I understand why you would feel uncomfortable sharing something so personal, but I'm sure your coworkers wouldn't judge you. If they work in a mental hospital, they definitely are sympathetic to people who are going through a tough time. Receiving mental health treatment when you need it is a good thing, not something to be ashamed of.