True, but at this point joining the EU pretty much involves consenting to adopt the Euro. Only Denmark and the UK have an opt-out afaik, all the other EU countries are obliged to join
The main thing is that you can't tax by printing currency anymore which is a really easy way to levy taxes. That means you have to have an efficient taxation apparatus, tax the rich and poor equitably and be very careful with borrowing.
Greek's problem is that it isn't particularly strong in any of those and thus isn't really suited for being part of the Eurozone yet.
This I think is the main difference with the USA. The USA also has richer and poorer stats, but because of the shared language, history and culture, people in both the richer and the poorer states feel like a US citizen.
They just exercise some creativity by making sure they don't meet the requirements.