I second using your union.
As for applying for EI, did you work while you were in school? If so you might be eligible for EI medical benefits, which are different than regular benefits if you lose your job. You can get paid up to four months. I had to use these last year when I had time off for a medical reason. You could ask your union about this as well. It's important to apply for EI promptly, ideally as soon as you stop working, if you delay too long you aren't eligible any longer. Definitely involve your union. I am sorry you are dealing with this on top of being a new grad, how stressful.
I work in psych. Unfortunately unfounded stigma against people with mental illness is a real thing. We would hope all health care professionals would be educated and not stigmatize, but unfortunately it is not always the case. On the other hand, some nurses are very accepting and supportive. I have had patients ask me, so, how and when should I tell people I have a mental illness? It can be a unclear decision when to disclose. In general I tell patients, once you tell someone, you can't untell them. If you're unsure of telling someone, you can always wait and see and tell them later. I tell patients, you have nothing to be ashamed of, and stigma is not your fault. If people are judgmental, it is a reflection on them, not you. That said, I say to them, your health information is your private business and you don't owe people to share your private business.
As for nursing license, I don't know how that works in Alberta. In my province the nurses college has an early intervention program. The idea is for nurses who have mental illness or addiction to seek assistance from the college so they can help you address it proactively, before it impacts your practice. I would ask your union about this as well, maybe Alberta has a similar program? In my province, fitness to practice questions are between you and your doctor. You are not obliged to share your private health information with your employer (diagnosis, etc), only to share if you are able to work or not, and if you need any modifications to your job for health reasons.
Some people get judgmental when they hear the term "bipolar", which can be unfair. A psychiatrist I've worked with has said, you know there are actually many successful, professional people who have bipolar, and are successfully managed as outpatients. Even people who are nurses and doctors, professors and lawyers. It can be done. It takes strength to admit when you have a mental illness and need support, and to get help. That is very tough when you are also a new grad, and I applaud you for being honest with yourself and getting medical help.
As for your job, wow that sounds like a stressful day, even for an experienced nurse. Try not to beat yourself up. Again, go to your union. All the best to you and I hope this is resolved quickly for you.