Would Brexit stop the flow of doctors and patients between EU countries?

The European Health Insurance Card is for tourists visiting another EU state. For employment and retirement purposes, the A1 and S1 forms are relevant. For people taking statutory retirement and moving to another member state, the S1 form advises the new country of residence that their healthcare costs will be paid by the country of state pension (e.g. residency in France, UK pension). This does not apply to people moving to another EU state before they take the statutory pension. This is a general right applying to everyone in the EU under these circumstances. The A1 is required if you go to work into another member state and fall ill; technically, a UK person running a workhop for pay in Germany would require the A1 form, not the EHIC should they fall ill.

The consequences of an exit vote are likely these:
1. The UK would need formal deals with each EU state (health policy is not an EU competency, but there are general regulations on free movement and access to healthcare, partly from European Court of Justice rulings, most of which the UK government has been on the losing side).
2. People travelling in the EU after an exit will require medical travel insurance to pay their bills as the A1 would not apply. They would likely need permission to work, too. This will increase business costs. Working in another EU state will probably require some sort of work permit; without a work permit, access to statutory healthcare is not available.
3. People taking their UK pension and retiring to another EU state would not get the S1 benefits, but need private medical insurance, as access to the statutory system would be closed off. As they would be "immigrants" they would need a work permit to work if they wanted to augment their income, but this would come with social charges. For many people choosing retirement in other EU states, this may be enough to change their minds and they may stay in the UK; others may move back to the UK rather than buy private insurance cover, especially if their pensions become capped through Brexit and likely taxed by their host country.

I doubt, though, that there would be an easy case for reciprocity as cross-border healthcare has been a difficult issue for the UK government to accept. Within the EU, cross-border care especially on border frontiers is more common as is the social insurance model which facilitates access to care through portability of insurance entitlements, quite different from the UK's system of entitlement based on residency. People who have lived in social insurance system countries accept eligibility criteria, insurance payments, co-payments and reimbursement of costs as the norm, (with little impact on equity of access to very high quality care).

As for emergency care when travelling, well it just wouldn't be there for UK travellers. Costs such as these are often subrogated by insurers who seek to recover costs incurred, while the EU system does not permit this. Wouid the NHS seek to subrogate costs incurred? Unlikely as UK citizens travelling in the post-exit would not have any NHS cover and would be at full risk of costs.

People perhaps do not fully appreciate the benefits of portability of healthcare entitlements within the EU. When they are gone, the replacement will lead to increased business and personal costs for medical travel insurance (which is substantially inferior to the EU benefits regime). There could be wider implications for public health surveillance, too.

Charging UK citizens in EU states for healthcare would be a straightforward procedure, given the social insurance regime. Premium costs would be in the hundreds of euros per month for cover per person. The UK's £200 charge is hardly comparable and conceals that the true costs are buried in the tax system.

Reference
Social security schemes and free movement of persons: Basic Regulation, 1408-71 as amended. There is an array of related legislation, but this is the basic regulation from which all else flows.