Time I live in Belgium, which iirc, is an EU member State.
UK citizens have been given guidance that they will be allowed to stay and work, but may have to get a visa, or look at citizenship from 2021.
Healthcare will remain in place, as long as you are in either an employers scheme (as dh and I are); a mutualiteit, or you pay for your own private healthcare. Those things are not nationality dependent as Belgium has huge amounts of international workers here, many of whom are not from EU member states, but places like the US, Canada, Aus, NZ, Turkey etc.
To say that any Brit living in an EU member state will become illegal overnight is not true. Yes, you have to find out what is going on, but there is plenty of stuff published by the FCO, the local embassies etc for advice and guidance. The only thing we did was to change our driving licences prior to the supposed exit date in March, as we are here until mid October, when we move back to the UK. Everything else was staying the same according to the Belgian authorities.
Which member states will make it illegal for Brits to be there?