I'm sorry you are going through this.
First, give yourself permission to be upset for a bit. Getting fired is a big, emotional, thing. It's totally understandable that you'd cry.
Given the circumstances here, it may be worth consulting with an employment lawyer. While nothing here is clearly illegal, a lawyer might notice something I don't. And whenever you aren't given a real reason for being fired, that can be used as leverage. So you might be able to negotiate severance or something similar.
Your list of filing for unemployment, updating your resume, and applying for jobs is exactly right. If health insurance was through your employer, you will want to look into either COBRA coverage or getting coverage through healthcare.gov marketplace. But certainly take the long weekend, at a minimum, to relax, gather yourself, and try to get into a better headspace.