The result would be catastrophic. The almost-reserve currency would collapse which would produce a medium sized depression, perhaps same with with 2008-now, or even larger, since don't forget, that one was produced from a housing bubble existing in only a part of the american economy; imagine what would happen if almost the full size of the economy (Europe) would collapse, even if Europe isn't as much "connected".
But reality here is, there's no chance to that. The real reason you hear those rumors is that America (along with minor partners like the British Sterling) want to bring down the Euro for medium-term benefit. e.g. Several economists get on Bloomberg announcing they are short selling the Euro.
Irony is, all this is helping the Euro since selling and short-selling and selling and short-selling helps massively its liquidity. It's like several nay sayers actually making a politician famous with their spite.