An egg can easily be stored for 3-4 weeks unrefridgerated (unless you wash them with chemicals like often happens in the US, then they need to be kept in the fridge for a shorter period of time). European food law puts the maximum consume-before date of eggs at 28 days, but usually you can keep them for a while longer.

If you don't reserve cargo space in advance, shipping across the Atlantic can easily take 20-40 days, depending on how annoying customs is being. The ship itself can be there in 9 days, weather permitting. Most ships also make stops at islands and other countries, so shipping time kind of depends on how direct the route can be.

If you're sure the eggs will be used within a reasonable time frame (a week), and you pre-arrange the transport ahead of time, you can have the eggs on US soil for a week before they spoil. If you keep them refrigerated during transport probably longer.

Other options also exist. For instance, one could pre-crack the eggs in Denmark, collect the yolk/whites/scales, and transport them in dehydrated form to the US. That's good enough for bakeries and factories, which would no longer need to buy eggs from the local markets, relieving pressure for non-industrial consumption.

If you refrigerate unwashed European eggs, they will last multiple months (I have kept eggs for more then 2 months), way past the expiration date on the packaging (I know because I do it constantly).