The problem is only just starting. There is a huge gap between those who run the EU and its member states, and the citizens of those states. Even Germans are 50/50 about the EU and the French are about 60/40 against. The Greeks must loathe it. The only people who love it are the poor countries who've done well out of it and the criminals who milk the corrupt and inefficient system.
When we joined the EU in 1973, Greece, Portugal and Spain were military dictatorships. It wouldn't surprise me if the desire for an EU army was to prevent something similar in the future.