I'm so sorry this happened at all, that it happened at this time of year, and that it happened in this unprofessional manner. As for the legality of it, well, as others have noted, if you are an at-will employee, then it likely is legal to fire you this way or any way. Unprofessional, yes, but unfortunately probably legal.
The question about legality makes me wonder if you are thinking of challenging the firing -- but if you are, why? If you challenge it, and you happen to win, you will have incurred a lot of cost in order to keep working in a situation that clearly isn't positive for you. Unfortunately too you probably have lost any chance of a good reference for another job teaching dance, and I hope you have a good reference from a past job -- someone you can turn to who will provide positive references when you apply elsewhere. But be aware that anyone looking to hire you has the right to ask what your last job was and why you left it. Or in this case, how it left you. If you're applying at a dance studio I would hope they'd understand that sometimes teaching styles clash and teachers move on because of that. Maybe you can leave it at that.
I read your past post about the boss who communicated with you by text (lousy management style) and the other teacher who is permitted to yell at students, etc. It sounds like a dysfunctional studio. The fact you were told never to correct one student is a huge red flag that the owner is more concerned about keeping some parent artificially happy than about actually teachiing a child dance. It says that the owner is interested just in keeping that family's money coming in and also is inept at handling parents who have an issue with something. I would be appalled if I thought a teacher were not correcting my daughter when it is needed (though I would be equally appalled if that teacher corrected by yelling or being cruel).
You might want to consider setting up as a dance teacher on your own; a person around here teaches dance in her home (she fitted out a studio in her basement) and isn't tied to any studio. If you do teach on your own, be sure to keep studying as well and taking master classes -- that will keep you sharp, get your name out there in the professional dance world, and make parents likelier to hire you, when they learn you take the time to keep training!