The EU will continue because people and goods need to be able to travel freely and do business throughout the region.
The Euro should be abandoned and sooner rather than later. The problem of creating a United States of Europe is that they are not actually a single country. Culture and predominate language limits the mobility of people in a way that is not a problem in the US.
Housing Boom in Spain and Florida?, In the US, Tennesseans and Ohioans move down and keep wages in check, but in Europe I doubt the Dutch and Austrians are moving to Spain. Economy poor in Greece and Mississippi?, move to Virginia or Kansas here but Are Greeks moving to Germany and France?
Economists like Paul Krugman warned of this from the very beginning, arguing that the system is not equipped to deal with asymmetrical shocks. Disciplined countries like Germany are going to have prop up their deadbeat brothers-in-law in Spain, Greece, and Italy or else put those countries through a long period of grinding deflation. Either way, one side or the other is going to revolt. The best solution long term is to let each country have their own currency so they can inflate their way out of this mess.
Whether it was pride or forethought, Britain staying out of the Euro was the right decision.