Longtime euro-critic, Paul Krugman tells Cyprus, "Leave the euro. Now." http://krugman.blogs.nytimes.com/2013/03/26/cyprus-seriously . He recognizes that it will probably not do so, but reasonably invoking the deep recession of neighboring Greece, he argues that they should leave before they end up like Greece. The question then arises, why are they not likely to do so, and for that matter, why has not Greece done so?
A lot of it of course is some sort of desire to "belong to Europe" and all that, which strongly influences both Greece and Cyprus, and continues to attract such possible joiners as Poland, which Krugman accurately notes did better than any other nation in Europe while not being in the euro during the Great Recession. Sweden also did well staying out, atlhough the UK is not such a great example. But, I think that there may be an economic fear that while probably overblown is not totally irrational. It is the fear of a possible major collapse in living standards from the likely massive devaluation that would arise if they were to get out (which as Krugman also notes, would not be all that easy to do).
The fact is that Cyprus is a very open economy, with imports running about 1/3 of GDP. A very major drop in the value of a new Cypriot currency would sharply increase the cost of living for Cypriots, and while the unemployment rate would certainly rise a lot, the entire citizenry would experience the potentially sharp decline in the standard of living. While this devaluation might make it easier for Cyprus to recover several years down the road, that recovery would indeed be several years down the road, and in the meantime there would be a lot of pain for the entire citizenry that will not happen if they stay with the euro.
The poster boy for this fear might be Argentina in 2001. The inflation rate went from actually negative just before the unpegging from the dollar to as high as 44% in 2002. The value of the peso fell to about 1/4 of what it was before, and while imports were only 22% of GDP, there was a massive short-term decline in living standards, with many middle class people at least temporarily thrown into poverty. That Cyprus would be moving off a long-in-place peg in a crisis environment suggests that it could experience a devaluation as bad as Argentina's or even worse.
OTOH, there is Iceland, a fave of Krugman's. Now they were floating, so not delinking from a longstanding peg. But in some ways they were arguably more vulnerable than Argentina or Cyprus, with imports running at 40-50% of GDP. After the 2008-09 financial collapse and banking crisis there, with many similarities to the Cyprus situation, their currency fell by more than half against both the dollar and the euro, but not by nearly as much as the Argentine peso fell. Inflation spiked from only around 2% to over 12% for two years in a row. They also had a recession, with the unemployment rate doubling, although that does not look all that much worse than what has happened in quite a few other European countries, and certainly not as bad as in Greece. But then, just as Argentina recovered after several years, so has Iceland's situation improved noticeably, although not all the way back to where it was before the crash.
Curiously, the more likely countries to see voters support an exit from the euro might well be some of the larger ones whose exit would be far more damaging to the euro itself (although some might disagree with that and say "good riddance"). The obvious case is Italy, where a majority of voters just supported candidates who at least criticized austerity policies imposed from outside, even if not necessarily supporting an exit from the euro (Grillo does, but Berlusconi has not so far). But such countries would probably not be hit as hard as Cyprus or Iceland or even Argentina for the simple reason that imports are smaller compared to GDP than in those smaller economies. A large devaluation, and it might not be as large anyway, would not impact the immediate cost of living as severely as happened in either Argentina or Iceland, or probably what would happen in either Greece or Cyprus.