COVID-19: some EU countries closed borders for foreigners - are EU citizens foreigners?

Yes, EU citizens are foreigners. The Czech regulations now only allow Czech citizens, foreigners with temporary residence over 90 days and permanent residents to enter the country.

Graphics by the Ministry of the Interior of CR

Details can be found in this document.

From midnight from Sunday to Monday, the ban to entry is extended to all foreigners, not only from high risk epidemic countries. Foreigners residing in the Czech Republic with temporary residence over 90 days or permanent residence are excluded from this measure.

1. The prohibition of entry, as referred in point I. of the Resolution, does not apply for:

Czech citizens or foreigners with temporary or permanent residence permit in the Czech Republic, who are demonstrably regularly crossing internal borders, i.e. cross-border workers working within 100 km from the state border,

citizens of Germany, Austria, Poland and Slovakia who are demonstrably regularly crossing internal borders, i.e. cross-border workers working within 100 km from the state border,

EU citizen transiting through the Czech Republic territory to his/her home country – this evidence of transit must be accompanied with signed declaration of honour,

foreigner with residence permit in another EU Member State transiting through the Czech Republic territory – this evidence of transit must be accompanied with signed declaration of honour.

To answer another part of your question (and comments) that doesn't seem to have been addressed directly: there is a clear difference between being an EU citizen and having residence. In fact, in normal times, your right to enter another EU country for a short-term visit is stronger than your right to take up residence in the same country. After some time, EU citizens even become “permanent residents”, being granted additional rights.

This is first and foremost a material fact, not necessarily tied to a specific document (even if it's typically possible to request one). For example, if you have a job and a home in the country and spend most of your time there, you're a resident. In practice, showing health insurance cards, bank cards, public transit passes, etc. all go towards establishing that even if you don't have formal proof or a specific permit or registration.

Obviously, in the rush to fight the outbreak, I doubt all the restrictions have been carefully crafted to take such situations into account but some countries have indicated that residents are welcome to return to their homes. This should in principle cover third-country nationals who require a permit and EU citizens who do not. Incidentally, that's not your situation but many countries also make an exception for cross-border workers.

I am not aware of any EU country that has put travel restrictions in place and makes no exception whatsoever for (some) EU citizens. Conversely, I don't know any that would grant non-resident EU citizens the same rights as their own nationals.

I think we have to expect more incomplete and poorly translated information bulletins during the next weeks.

If we look into the wording, as the regulation is currently published by the Polish Government, the exception applies to:

• persons who have the right of permanent or temporary residence in the Republic of Poland or a work permit

A 'residence permit' (as in a physical document) is not necessary. The 'right of residence', which EU/EEA citizens have, is adequate, as of now, to be allowed to enter Poland.