You may have a claim under the National Labor Relations Act if the employer attempts to universally ban the wearing of all union insignia, even in a nonunion workplace. Employers are allowed to set neutral policies which prohibit certain types of clothing, such as t-shirts with union logos if the employer bans all t-shirts, if the employer enforces the policy uniformly.
However, several courts have determined that employees have the right to wear union buttons and pins to work, with two exceptions:
if wearing these items creates a safety hazard or,
in the case of workers with public contact, if the employees consistently are required to wear uniforms without buttons and pins.