I am actually curious how you would go about proving that you are a citizen of the EU. I am sure you need some form of identification to board an international flight even though it might be within the Schengen area.
So, from experience squatting in the Netherlands... nobody bothers to check, unless you look like you might be from Eastern Europe or outside Europe, at which point they throw you in a foreigner detention centre until you can prove your identity or they can deport you to wherever they think you might be from.
My experience driving through the EU is that barely anybody cares. You may be stopped at internal borders if your car plates don't match a Schengen-area country, but that's it.
You'll need your ID to stay in a hotel/AirBnB though. So unless you squat or camp illegally, better have it with you.