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Mark, the basic problem of the euro is debt-related.
If you have a house and a family income, and a fixed household budget, but a debt that is superior to this, and find you cannot keep up with repayments of the debt, then it's time to think of change, of finding a new house which is not yours but rented out to you.
In fact, your creditors will demand their owed money and you'll have to undergo a case of bankruptcy in court.
With a state, it is as follows: if you have a sovereign debt, and a fixed state budget, you have to allign the budget to keep up with repayments of the national debt.
If, however, the debt keeps rising, and you have to continue cutting into the budget to keep up with the repayments on debt, including the interest, and at some point realise that the State needs a bailout (a new loan that must be repaid with the interest), and if there are no prospects of getting out of this vicious circle (for example there are no oil-wells and no gold mines to be discovered within the sovereign territory), then you have to think of change.
If the eurozone countries do not start thinking of changing their situation, while at the same time having no prospects of financial recovery, then they are essentially wasting their time.
It's hard to see what example could be learned from the history of the USA in solving the financial problems of many a eurozone country.
In the timespan of history of the USA which you refer to, the national territory of America increased by tens of times, new states were being founded on a yearly basis, and tens of million of people were emigrating to America.
Gold and oil were discovered in the USA, and mines were dug for minerals, things which the eurozone states cannot dream of having.
