The Currency is completely separate -- you do know that many of the European union states do NOT have the Euro, right?
And even if the Euro fails, the EU is a separate entity
Finally, the lack of borders is not quite how you are seeing it -- look at Germany itself, it is a federation with the individual LANDen like Bavaria having near independence. Or the individual cantons in Switzerland. The EU is similar with a far far weaker center than even Switzerland.
I had completely forgotten Switzerland. Thank you for reminding me about how the Swiss are worried to death for the future of their country since they never got around to join the EU. Foolish Swiss! They are doomed now!