There is no hospital in Clayton, but there are small clinics, which according to my friends who live there, are really good. There are also dentists in Clayton.
The closest private hospital you can go to if you live in Clayton would be Punta Pacifica and Punta Paitilla, which are both just 15 to 20 minutes away from Clayton by car. There are also a couple of state hospitals that you can go to, but the private hospitals are really good. The Punta Pacifica Hospital is a world class hospital that is affiliated with Johns Hopkins and it is very famous in Panama.