If you are an American citizen you do not need permission to come home.
Is it illegal for an American to cross the border legally but cross back to the USA without permission?
When I was about 10 years old and living in Montana, we had been visiting some friends in Canada. Upon returning to the US, my father inadvertently drove through the Canadian checkpoint and directly into the US checkpoint. (They were very close together.)
It was an honest mistake, and we were shortly on our way home, but it caused a considerable commotion in the short term.
I was too young to know if the primary problem was with the Canadian border officials or the American side as well.
Technically ... If an American citizen doesn't have a valid FEDERAL-accepted form of ID, they can't enter the US. (Nor can they even travel by air within the US.)
Of course, since millions of "undocumented workers" have crossed the US borders and not been turned away - how would the federal government keep its own citizens out? LOL