Military Family Stability Act This bill directs the Department of Defense (DOD) to prescribe regulations that permit an eligible member of the Armed Forces who is undergoing a permanent change of station within the United States to request specified housing treatment during the period from 180 days before to 180 days after such change. An "eligible member" is a member who: has a spouse who is gainfully employed or enrolled in a degree, certificate, or license granting program at the beginning of such period; has one or more dependents attending an elementary or secondary school at the beginning of such period; has one or more dependents enrolled in the Exceptional Family Member Program; or is caring for an immediate family member with a chronic or long-term illness at the beginning of such period. If a spouse or other dependent of a member whose request for such housing treatment is approved: (1) resides in government-owned or government-leased housing at the beginning of such period, he or she may continue to reside there during such period; and (2) relocates at a time different from the member, the member may be assigned to housing intended for members without a spouse or dependent until the member's detachment date or the spouse or other dependent's arrival date, if such housing is available without displacing a member without a spouse or dependent. (The bill specifies alternatives for the basic housing allowance to be provided in such case.) The Government Accountability Office shall report to Congress on potential DOD actions to enhance the well-being of military families undergoing a permanent change of station.