Both my wife and I are US citizens. Both of our children were born overseas when I was stationed in Germany. I find it hard to believe that both are not natural born citizens.
Were your children issued Certificates of Naturalization? If they weren’t, they’re considered to be Citizens of the United States at Birth.
Definition C in the US Code for Nationals and Citizens of the United States At Birth seems to apply to your family: “(c) a person born outside of the United States and its outlying possessions of parents both of whom are citizens of the United States and one of whom has had a residence in the United States or one of its outlying possessions, prior to the birth of such person;”
According to Vattel, they are.
His argument makes perfect sense to me.