Legal Question in Entertainment Law in California

You might not be able to retain exclusive rights in the phrase if you permitted the school to use it on equipment and t-shirts without having some sort of written agreement with them. They may, rightly, consider your contribution a gift that they can keep and use as they want.

A trademark is something that is used in commerce and is associated with a particular business. If you don't have a business in which you regularly use the mark, it is unlikely you would be able to register it as a trademark. If the school is using the phrase, it is more likely they would be able to register it as a trademark.