The last time I used an EAP, they connected me to a therapist and that was it. I was having panic attacks due to a then recent incident with my mother and was in a lot of distress.

I've heard some negative things about EAPs from others, but that wasn't my experience. In fact, my experience was that HR and the job had no real interaction with it. It definitely helped me at the time, and no, I didn't lose my job. I did end up leaving that job for a promotion elsewhere, but I also have to say I worked at a place that had pretty great HR philosophy (that I have yet to really experience anywhere else - this was also the HR program that had set up a health insurance program and leave program for employees who weren't eligible for the main program yet when they were new hires, and I actually did end up with a major illness within my first month of work, and it helped me keep my job and pay the healthcare bills).

My understanding is that most HR departments don't really care much about the health and wellbeing of employees, so I'd gauge what the philosophy and behavior of your HR is. That said, if you have access to EAP, it's supposed to be free, so use it. It's not like a therapist can tell your employer what you talk about; it's all confidential.