It's distinctly possible that might be the case, Valery.
I believe Italy, Spain and France, as longer standing EU members may have a stronger preference to EU countries of shorter membership such as Poland, Hungary, Slovakia or the Czech Republic.
Some countries refuse outright to hire non EU citizens for teaching positions, I'm told that's the case in the BENELUX (Belgium, Netherlands and Luxembourg) region.
Your best bet is to check the embassy websites of the countries you want to teach in and see if they do have some stipulation written into their labour code that bars non EU citizens from teaching there. Most countries will at least have a basic outline of their labour code as it applies to foreigners somewhere on the web site.
If you can't find anything that specifically rules out non EU citizens from teaching, fire out your resumes. If the information is not clear, call the embassy and ask some questions about the matter.
One of the primary reasons that an EU country would bar a non EU citizen from teaching is that there is virtually no bureaucracy (or comparatively little) when hiring an EU citizen in comparison to a non EU citizen.