The jpoc guide to health cover when you work abroad
The E111 and E128 health care forms
The EU & EEA have created a scheme that allows their citizens to receive health treatment outside their normal country of residence on terms similar to those that they would receive if they were a national of the country in which they are seeking treatment.
These schemes are operated by means of an assortment of forms that you can get in your home country. Essentially, a UK citizen can obtain such a form in the UK and use it as a sort of health passport when outside the UK.
The two most common forms are the E111 and the E128 and these are the only forms that are of relevance to British citizens travelling around the EU and EEA.
One important thing to bear in mind though is that the scheme is not designed to address the health care needs of people leaving their normal country of residence in order to take up work in another country.
It is possible to structure your affairs so that you can obtain one of these forms. This will almost certainly be OK if some official says that you must show evidence of health cover in order to register in the country in which you will work. That does not mean that you will actually receive the cover if you need treatment. You may end up with a large bill later.