The remain campaign leaflet correctly says that UK citizens can get free (or cheaper) healthcare while travelling in the EU if they have a European Health Card (the E111?). I say cheaper because (I think) it entitles you to healthcare at the going rate for citizens in that country which is not free everywhere. Something worth checking before you travel as you might want extra insurance.
So the implied concern is that we lose that benefit if we leave. Well like most of these things it is not a slam dunk. The card also gives you access to healthcare in the European Economic Area (EEA) which includes the EU and a few other countries. The card also works in Switzerland which isn’t in the EU or the EEA. So it is possible to keep the system if we left the EU and there is a precedent for doing this.