The Scout Master, an Assistant Scout Master or an Eagle Scout can sign-off on rank requirements. So I can't sign my son's book?

No you cannot. Only the Scout Master, an Assistant Scout Master or one of our Eagle Scouts can sign-off on rank requirements. This is a check and balance in Boy Scouts. For instance, we have Adult Leaders who are Committee Members. They cannot sign-off on rank requirements either, because they will be the ones conducting the Board of Review. And conversely, the Scout Master and Assistant Scout Masters are not allowed to sit on a Board of Review.