You should definitely go ahead and file your unemployment. Of course you'll be looking for another job while you are filing, getting the issues worked out, seeing if you can be approved. It will take a long time, but if you file as soon as you're out of work, that's weeks you could possibly be back paid for.
You didn't say how long you had worked for this employer. But in order to be able to draw unemployment, you must have worked and have wages for covered employers during about the last 18 months to two years. You will file. They will ask certain basic questions on your claim.
What is the reason you are no longer working at your job? What is the reason you were told you were terminated? What happened on the last day you worked? Had you had any prior warnings or write ups concerning this issue before you were fired? What did you do to try to work this problem out?
If you were given a warning about the piercings, and then several weeks later were called by corporate and told to turn your keys in, with no call from your supervisor, was there a precipitating incident? Did you, by any chance, show back up at work with your facial piercings in? Were you supposed to be texting everyone and discussing the issue while at work? Did you have any reason to believe you did or did you do anything that caused your firing beyond the warnings?
Just give the unemployment system the facts. They will then contact the employer to get their version of why you were terminated, and they will make a decision whether or not you get to draw benefits. You will be certifying for each week during this time that you are not working. If the decision goes in your favor, you'll be back paid and will then begin receiving checks. Either party that does not prevail can appeal with a hearing.
In order to terminate you and keep you from drawing unemployment, the company will have to prove to the unemployment system that they had a valid misconduct reason to terminate you. They should have a trail of warnings and progressive discipline, where you were given the opportunity to change your behavior and save your job, and you chose not to.
But firing you any way they want to is legal. They can invade your privacy pretty much if you are using their electronic systems. They can claim they called you when they didn't, legally, and fire you without an exit interview or without giving you any warning or reason. But that does help your chances of getting unemployment. File the claim, there's no downside to doing it, even if you ultimately don't get approved.