A number of EU member states can form a political union between them (outside the EU Treaty) and continue to be members of the EU but as a single member state!!
So if let's say all 17 Eurozone members merged into one (federal country) then this new country, let's call it "Federal Republic of Europe" (FRE) could be a member of the EU instead of the 17, in other words an EU with 11 (10+1) member states.
Of course this new "country", FRE, would have its own publlc servants and its own institutions which would probably be staffed by former members of the former national administrations.
If indeed this country consisted of all members of the Eurozone, then that would go a long way towards solving the Euro crisis. It would have a Prime Minister elected in the same or similar way the German PM is, have a federal tax system and a federal "IRS" to administer it, maybe its own army, and of course FRE-wide laws. Its own bonds, etc.
Now, if this new country did not match the current Eurozone in membership, that would not help solve the Euro crisis, ie the Euro would not gain a country, but it would still go a long way towards the destination of European integration. Over the years and decades more European counties, EU members or not, would merge into this FRE country.
That would not be good new for the EU per se, but the EU (much like the EEC) is but a vehicle for European integration. What matters is European integration not the EU per se.
This FRE solution could work irrespective of the future of the Euro, even providing alternative impetus in Euro's worst case scenario (the FRE would not necessarily have a single currency at least at first).
There are other solutions too of course.