The connection between government debts and unemployment is illusionary.
I'm sure that somehow a case could somehow be made that there is some relationship, but not that one causes the other.
In general, you are correct. Unemployment is initially more likely to be the cause of higher government debt load than the result of such a debt load. But what's happened in Greece is that the government has been forced to take such significant austerity measures that it has laid off large numbers of government workers and ended large numbers of government contracts, which has driven up the unemployment rate.
In the past, Greece would have been able to deflate its currency and this would have significantly eased its debt crisis. It can't do that now because it's a member of the EU.
The suggestion that "debt" is dragging down the stronger EU economies is correct, but only in a tangental way. The connection to Greece, Spain and Ireland is what's dragging down those economies, not their own debt. This is, quite frankly, the EU's fault. The stronger EU countries linked their economies with weaker economies and then failed to take any significant regulatory steps to ensure the weaker countries were fiscally responsible. To resort to the overly simplistic analogies that this thread favors, it's as if Germany and France took out a joint credit card with Greece, failed to notice that Greece was spending way in excess of its means, and now Germany has the unsavory choice of paying Greece's bill or facing some consequences.
The U.S. is a completely different animal for a variety of ways. We are a reserve currency. We have almost 0 percent borrowing costs. We have a variety of ways to raise revenue that we have not tapped. We have not attempted any significant entitlement reform or addressed our runaway military spending (more per year than any other five countries combined).
The biggest difference between the simplistic analogy above and ... well ... reality is that the example ignores the fact that the U.S. has the ability to raise its "salary" or revenues at any time by raising taxes. This is exactly what will happen eventually and the longer it takes to happen, the higher the taxes are going to be. A wise person would be preparing for this in a variety of ways.