Short Answer: No.
A belly-button (clinically called the umbilicus) is a scar on the abdomen of placental mammals at the attachment site of the umbilical cord. Simply put, it is a remnant of your birth.
Looking at the creation account in Genesis, Adam and Eve were not born, but created. Therefore, they did not have belly-buttons.
(I bet it must have been an interesting conversation piece with the kids, huh?)