Employees can almost always wear union logos
The National Labor Relations Act provides powerful support for employees who want to join a labor union. It generally permits employees to wear a union logo on their clothing at work. Punishing employees for doing so will often result in the filing of unfair labor practices charges.
Recent case: North Memorial Health Care, in the Twin Cities area, is a union workplace. Union members regularly meet with union representatives in the hospital cafeteria.
During a labor dispute with management, several employees were punished for wearing union logos. The National Labor Relations Board said that was an unfair labor practice and the hospital appealed.
Now the 8th Circuit Court of Appeals has sided with the NLRB and upheld the charges. However, the court did say that the hospital could prevent nonemployees from wearing the logo. (North Memorial Health v. NLRB, 8th Cir., 2017)