YOu feel betrayed because you have been betrayed. They never should have referred you to EAP without talking to you about it first, never mind talking about it to your new boss first. I really think that it is never a good idea to trust people who can have a negative impact on our life , such as work, with our psychological problems. You are much better off discussing them with a friend, someone who has the same problems as you or a professional. Dont let these two people turn you off totally to trust. It is a lesson however in not leaving yourself vulnerable to repercussions because you trust the wrong people. Talk to people who can only help you not use the information against you to hurt you .
Do you have to go to EAP now that you have been referred? If you do go in with your head up and tell them you were just having a bad day and that you can do your job. That you confided in people and they exaggerated what you were saying. if you have to go for counseling for a period of time, find your own place to go for counseling not the employers counseling center. I dont trust them at all and know of violations in confidentiality. YOu are not in a great spot right now but it doesnt have to bring you down. I wouldnt even bother talking to the other two about it at all since no matter what you say they will view it as part of your problem. Just move on and talk to them as little as possible in the future.