A scout finishes all his requirements the day before his 18th birthday......except for this EBOR. Well, Sonny, you had 7 years to figure this out, you knew it was coming, and the rules are the rules. Too bad.
I would just remind the boy that life is not fair and the rules are the rules. He's over 18 years of age anyway so he's not a registered Boy Scout and one doesn't need to handle this situation with kid gloves.
Life can change in an instant,..... Happy Birthday.
But the Guide to Advancement specifically says the EBOR can take place after the 18th birthday. (With no permission from anybody within 3 months, or with council permission between 3 and 6 months, or with national approval after that.) Some Eagle candidates rely on the fact that they can do the EBOR after their birthday. (Too many, and my son ended up being one of them, and yes, technically he was not a registered member of the BSA at the time he had his EBOR. Two of our troop's last three Eagles weren't either.)
There is nothing magic in that rule. The BSA could just as easily decide that the EBOR must take place before the Scout is 18, but I think they would have the change take effect at some future date after announcing it, like 6 to 12 months. It would effectively shorten the time the Scout has to complete the other requirements, and it would affect different Scouts differently depending on how EBOR's are scheduled in their council. (Which are two of the reasons it probably will not be changed.)