Here in Czech Republic politicians often claim that we cannot accept the Euro until 'the situation in Greece and other problematic Eurozone countries is resolved'. But given that joining the Euro doesn't incur any financial burdens due to the lack of a fiscal union, why would any problems in Greece threaten other Eurozone members? I can see how a Greek default might affect bond holders in other countries but that's unrelated to the currency itself.
The monetary impact of a default in Greece should be very small in the Euro given that the percentage of GPB of Greece is very small compared to the rest of the Eurozone.
However such default would mean that a large institution as the European Central Bank is unable to deal with a small problem like is a hypothetical default of the Greece government. This would lead to a downgrade of the qualification of all fund denominated in Euros such are the sovereign debt of the countries in the Eurozone.
This downgrade would mean less confidence in the ECB what is one scenario that would create a negative impact in the Euro.
What could be the consequences of such impact? Probably the Euro would lose some of it's value what is bad for the imports in the Eurozone. You have to pay more Euros for the same value of good or services purchased out of the Eurozone.
The impact of the exports would be positive as every good or service sold by countries in the Eurozone would turn out to be cheaper without any substantial change in the productive model in the UE.
In my opinion, keeping an Euro at a value in the range of 1.1$ 1.2$ is one of the current objectives of the ECB. Political leaders tend to emphasise whatever they can do to benefit the directives given by the ECB.
But given that joining the Euro doesn't incur any financial burdens due to the lack of a fiscal union
One problem is that a possible solution to the Greek crisis would be to create a fiscal union that bails out Greece. Until the crisis is resolved, a new country mind find itself joining just in time for a new tax.
A monetary option would be for the European Union central bank to bail out Greece through quantitative easing. Basically they would buy all the Greek debt and then make a deal forgiving some or all of it. This would have a devaluing effect on the euro. Other countries may not want to join a euro just before it devalues. Better to do it afterward.
There's also the general problem of joining right before the euro starts splitting. If Greece leaves, what happens to Portugal, Spain, etc. If Catalonia secedes from Spain, what happens to the euro there?
Borrowing costs could increase. Part of the problem is that people loaned money to Greece thinking that Grexit was impossible. If that turns out to be false, borrowing costs of other euro-denominated countries might go up. In fact, there are some signs of this, like rising bond yields.
Finally, all the above may be the excuses while the real reason is that they don't want to give up control of the currency. I.e. they don't want currency union, so they are looking for excuses. Or perhaps a mixture of reasons.