There are only little clinics in Clayton. They don’t really have big hospitals there. If you go to the clinic for all but the most serious or complex health issues, I don’t think there will be a problem, but you wouldn’t go for an emergency or something really bad. If you have an emergency, you have to go straight to a hospital.
If you just want to have a doctor’s appointment, you can go the next day. You just have to call and let them know that you are coming.
(Road in Clayton, previously an American base in Panama, pictured.)