My first therapy experience (as a client) was at the college where I was taking general psychology classes. They offered a limited number of session (it was 8) to anyone who cared to use some counseling. I didn't hear of anyone being turned down on the basis of the severity of their problem or for any other reason, but, as I said, a number of sessions was very limited - only 8. But that was about 15 years ago. I don't know how they do thigs now.

As to your records, your health information cannot be shared without your permission with anyone except in some extraordinary circumstances. However, technically, the file with your records belongs to the university and those who work at the univiersity health center or counseling center will have an access to it. But no one outside of that department will have that access, and certainly not your boss, nor would anyone from the counseling center share your information with outsiders. So, your boss definitely wouldn't know. If he gets your information somehow, that would mean someone broke the law, but I don't believe they would do that.