When you file a claim for unemployment benefits, it may help shed light on the subject for you. You will file, saying you were simply called in one day without any prior warnings or any idea what you did wrong and told that you are being fired for behaving inappropriately. As long as you have monetary eligibility for benefit (have been working for a covered employer during the last two years enough to have sufficient wages to set up a claim) then they will make an eligibility determination based upon your reason for separation. The employer will be contacted and asked why you were terminated.
There are two reasons for termination that will keep you for being approved for unemployment. These are job related misconduct of some type for which you were given sufficient warnings and indications that your behavior was unacceptable and if you continued with it, you would be fired. Then the employer would show that you did in fact continue the misbehavior, and this was the reason you were terminated.
Or else it can be gross misconduct, which is something so blatant and wrong that any reasonable person would have known they shouldn't do it, even once, and may be terminated for doing it even if they had no prior warnings. Setting the building or fire, punching out a co worker, stealing from the med carts, there are a number of this type offenses.
No, in order to terminate you legally, they do not have to have any proof you actually did something. But to keep you from receiving unemployment benefits, which cost the business money, they would have to show the unemployment system that they have reasonable proof you did what they think you did. A decision would have to be made on whether or not they had sufficient reason, or else you'd be granted unemployment benefits while looking for another job.
See where this would help you figure it all out? At least you'll have some idea why you were fired, unless, as they have done sometimes, the employer tells the unemployment system you have been let go for a non fault reason, in other words, chooses not to oppose your receiving benefits. Sometimes they just want rid of you, and just want you to go away without any issues or further contact, and they'll not object to your receiving benefits and will refuse to provide a reason why you were terminated. There's nothing illegal about them doing this, either.
Honestly, most people have some clue of what it is they may be accused of, or what may have been going on. But if you don't that still isn't illegal. They can fire you at will, and don't legally have to give you a reason.