Would I lose free access to their health services when I travel to Europe?

No!. Britain has reciprocal health arrangements with EU countries that have comparable national health services. e.g. Germany, France, Holland.

Why should existing reciprocal arrangements  change if the the UK was free and independent?

Many European countries simply do not have a comparable public health service even to EU members. EU citizens, including UK citizens have to pay or have private health insurance.

The current system does not work in Britain’s favour anyway.  Latest Dept of Health figures show:

The NHS paid £674 million to treat UK citizens in European countries.

European countries paid the NHS £50 million to treat EU citizens in the UK.

France received £150 million but paid the NHS only £6.7 million

Spain received £223 million but paid the NHS only £3.4 million

Germany was paid £25.9 million and paid the NHS only £2.2 million.

Labour MP John Mann said, “Sorting this scandal out would transform the financial situation of the NHS”. [viii]