Can job turn down a supervisor position because of a health condition

Generally, this kind of decision is unlawful.  Employers cannot discriminate against employees in promotion decisions on the basis of a disabling medical condition, which her fibromyalgia may be, depending upon how serious and debilitating it is.  Even if it's not, she may have a claim if the employer perceives her as disabled due to the fibromyalgia.  These cases are fact-specific, so she should call us to schedule a consultation.