Although I largely agree with Denise's answer, I'd like to add something important: if a doctor says that a mask is needed as a reasonable accommodation of the employee's medical condition, that creates protections under the ADA (and the Washington law equivalent of the ADA). So, best to get a doctor's note!
-Alex

The first step is to determine what is the hospital's policy regarding wearing a mask at all times while screening patients. Some hospital staff are required to wear mask, but that depends on the type of patients they are assigned. This is not the general rule for all hospital workers, it differs on a case by case basis.

The next step is to determine whether your family member had a written doctor's excuse for wearing the mask, and was that excuse communicated to the nurse manager. If your family member was wearing a mask as a protective measure against all patients, then such measure may not align with the hospital policy.

If the hospital policy prohibits a person from wearing a mask at all times, with some exceptions to the rule, then wearing a mask would be a violation of that policy, and the hospital could fire a staff if such actions rise to the level of termination. If your relative had a legitimate reason, and a doctor's note, that permits him/her to wear a mask while on duty, and such notice was communicated to the supervisor/ manager, then the hospital would not have a right to terminate employment.

When a defendant asks a court for a Summary Judgment, "The judgment sought shall be rendered forthwith if the pleadings, depositions, answers to interrogatories, and admissions on file, together with the affidavits, if any, show that there is no genuine issue as to any material fact and that the moving
party is entitled to a judgment as a matter of law."
http://www.courts.wa.gov/court_rules/pdf/CR/SUP_CR_56_00_00.pdf