As I noted below, the Eurozone and the European Union have substantially non-overlapping memberships at this point. And as Wolfgang Munchau argues in a persuasive-but-sadly-non-concise post, after the failure to reach an agreement with the United Kingdom it seems like saving the single currency will require killing the Union. Whatever is or isn't agreed to this week, it's clear that for the Euro to persist in anything resembling its current form really does require a path of "ever closer union." On the table right now are financial regulations and tight fiscal controls. But Germany is going to want to continue to push for labor market reforms and whether they like it or not something has to be done to rebalance the intra-Eurozone trade and financial flows. You can't do any of that without deploying the institutional apparatus in Brussels, but the United Kingdom isn't going to want any part of any of that. At the same time, it would make no sense for a continent sized currency-and-banking union to have its main financial center offshore in London.
You could imagine Sweden and Denmark decided to go all-in on this new tighter union or else following Norway to just have a free trade pact with a Union its aloof from. But the UK would definitely want out. And an EU without a UK would have a totally different character as a foreign policy entity. It would have less muscle, but quite possibly more coherence. (Alternatively, of course, this might just all be grandiose and unrealistic -- I vacillate between the view that saving the euro would require unthinkable steps and that ending would entail unthinkable consequences).