I don't think this is true. The AOO happens right before they leave my "reach". They have already moved before the AOO is triggered. With a 5 foot move they have moved almost entirely out of their sqaure they were in and are almost entirely inside the square they moved to. If you are going to up them inside one square it would be the one they are going to. It would also take almost 10 movement to move back to the center of the square they were in (almost 5 to move out, almost 5 to move back)
I can see the arguement that they are still within reach, but not the arguement that they are in their original square.
I also think he takes the booming blade damage because the wording of the booming blade spell is "target willingly moves 5 feet or more". Moving 5 feet moves him to the extent of my reach. An enemy that is exactly 5 feet away is exactly at the edge of my reach, the spell triggers at the same time as the AOO triggers.
If the spell said "target willingly moves more than 5 feet" then he would not take the damage, but "5 feet or more" would mean he does.