Well, you would think so, wouldn't you, and about 60% of all men between 18 and 60 left for north america when the war was about to start. But in every family, someone had to stay behind to keep the claim to the land. The germans were trying their best to germanize the area by moving germans onto the land. If everybody had left, the area would have been german today.
After WWII a plebescite was held, and the area returned to Denmark.
From what I understand, the "danish" soldiers caused a lot of trouble for the german army, I know especially one of my grandfathers story, and I don't think any of them shot at any british or french soldiers.
The experience thoroughly turned them off soldiering.