The EU itself has no power to do that. Each country has to temporarily re-introduce border control. They often do!

https://ec.europa.eu/home-affairs/what-we-do/policies/border...

It's not a power the EU has. Health care as well is just not a subject the EU has any role in.

Not so - there are many areas where the EU is involved: every EU citizen gets the same access to healthcare in each country as their own citizens, that's one of the rules of membership. Coordinated health research just to name another area...

Not the EU as an institution but single countries. I wouldn't be surprised if Austria closes their border if there are more cases in Northern Italy. Germany, Switzerland and France could also follow. I believe they'd still let freight through (maybe with tests for drivers) to not hurt the economy too much. Ironically, the refugee situation a few years back helped those countries as they now have the infrastructure to patrol borders better and built centers that could as well be used to isolate potentially infected people.

But I also wouldn't count on all of this happening too fast, there's still passenger flights arriving from mainland China in Europe.

Tourism, in all directions, is too important a factor that closing up borders is really on the table right now. Austria for instance, where I come from, will definitely NOT close the borders to Italy in the upcoming days. And in my personal opinion, there is really no reason to panic because of a media-hyped flu outbreak.

famous last words of a certain Hubei governor.

go read the treaties before commenting about this topic.