Years ago I did contact my EAP. For the most part it was a positive experience. If a supervisor is doing things correctly, they should work through the EAP when someone is depressed. For two years I was a supervisor and needed to work with someone who was depressed. When a supervisor works with EAP and has an open mind, things do work out much better. You remind me so much about my own experience at work. There came a time when my supervisor began to make my life hell and did everything he could to push me out. It destroyed my self confidence and made me extremely depressed. When someone is as depressed as you are now it is almost impossible to fight back. You may need to get involved with a more intensive program IOP etc to get the help you need. My heart goes out to you. I was in your same position three years ago.
PS you will NEVER be personna non grata here