Armies are not allowed to do any policing, without special approval from the host nation.
That's why armies and police are separate. Police protects the people from within, army protects the people from enemies from without. If you let armies do policing, the people tend to become the enemy.
EU army makes sense. We're already doing it. Belgium shares its navy with the Dutch, its jet fighter training with the French and has a shared army battalion with the Germans.
Most EU countries are too small (or have too small budgets) to have a worthwhile army, so why not share? We have similar goals: national security, peace keeping, etc.