There are small pharmacies and clinic in Mui Ne. Phan Thiet offers better facilities and large hospitals. For serious procedures, you may want to head to Saigon which is 3-4 hours away. You can purchase just about any medical supplies you could need it Phan Thiet. Dental is also available here.
Binh Thuan Red Cross
Address: 30-32 Ly Thuong Kiet, Phan Thiet