The jpoc guide to health cover when you work abroad
The E111 health care forms
The E111 is a form that you can take with you when you travel to other countries within the EU & EEA. Under certain circumstances, this form will entitle you to medical care in those countries but, it is for visits. It is not for working. If you are a permanent employee of a British company and you travel to (say) Paris for a one day sales presentation, an E111 is no good if you have a medical emergency in France. You were working there, albeit just for a day, and you are not covered. Technically, you are also not covered it you fly to Frankfurt for a job interview.
I often hear from British contractors working in Germany that they have a form E111 and so they do not need any medical insurance. Well, the answer is simple. If you are working in Germany, you cannot use an E111 to get medical treatment there on the NHS. It's that simple and so clear. According to the UK government regulations and the EU & EEA treaties, you are not entitled to cover.
There is another form, the E128 which is of use to people working outside their normal country of residence but it is generally not of relevance to IT contractors. However, if you do have one of those forms, you should also carry an E111. If you are working in Germany with an E128 form, then you will have no cover if you are taken ill in another EU or EEA country. You should then carry the E111 form for that cover.
Remember also that the E111 for is for emergency treatment only. If you suffer from some problem that would not have you in the A&E department in the UK, you are not entitled to free treatment with the E111.