Skiathos does not have a hospital , but a medical centre which only caters for basic medical problems. If necessary you will have to be transported to the mainland for anything serious. Make sure you have adequate Travel Insurance cover before travelling.
The medical centre is well signposted and is situated at the top end of town near to bus stop 4 in an area known as the Acropolis. There is a doctors surgery in Troulos.
It seems that the best place to go if you need to see a health professional is to the health centre at the Acropolis. (just down from bus stop 4). Whilst there are a few clinics in other resorts these are private and, as such, it is very unlikely that they will take your EHIC card.