This is absolute nonsense. How we organise our health service, and how we pay for it, is a matter for our own government.
But indirectly, the NHS and the health of British citizens both benefit from being part of the EU. In a letter to the Times on 4 April 2016 (not free), health professionals and researchers spoke of the benefits of continued EU membership to the NHS.
Most importantly, they debunked the myth that EU trade deals will jeopardise the NHS. In fact, EU law makes it clear that EU countries have the right to run nationalised public services, and no EU rules can change this. More details
In terms of staffing, with tens of thousands of nursing and doctor posts vacant, we cannot afford to risk losing the nurses and doctors from other EU countries that work here. Evidence
As for healthcare abroad, a mutual agreement between EU countries means that British citizens are entitled to emergency medical treatment when travelling in Europe. Evidence
If Britain were to leave the EU, UK nationals would no longer be automatically entitled to free emergency healthcare abroad, leaving travellers and holidaymakers £773m a year worse off. Britain Stronger in Europe calculates that the average British citizen claiming abroad received £6200 worth of free healthcare — something we could find ourselves paying for if the UK were not part of the EU. Evidence