Can I be fired from my job because I won’t cut my hair?

I'm afraid the you'll have to visit a barber. Legally, an employer can enforce rules about how employees look; this includes hairstyles. It is up to an employer's discretion as to what harstyles are or are not acceptable to it. This is especially true if the job involves interaction with the public. In customer service positions, an employer has a strong interest in just how its employees look. In employment "at will" relationships, an employer can hire/fire, reduce/increase salary/hours and otherwise impose job requirements as it sees fit.

Bottom line, unless your employer's actions violate its own company policy, or the terms of a union/employment contract, or is in some way constitutes legally actionable discrimination, it is breaking no laws in your case.

Note: Legally actionable discrimination is discrimination that results from a person's belonging to a legally "protected class". A protected class is one based on a person's inclusion in a group due to factors of race, religion, gender, age, disability, sexual orientation, national origin, etc.