Put it this way. Body is designed to Transport the Head, not the other way round. If the chin is Dropped, the Head dictates where the Body goes, it alters the Biomechanics of movement, alters Posture. I see it a Lot.
A good example is Rios. His chin is down. Clearly he doesn't have the best movement, but it's effective for him. Most boxers keep their chin tucked, but not necessarily down. I do have a problem with guys like Andy Lee, Wlad, etc who strictly keep their chins in but not down. It looks ridiculous and makes it hard to avoid a rush. It's good general practice but trying too hard to stick to it makes you fight like a robot and often forget to move your head.
The point he is making is not that you don't tuck your chin but the method by which you do it. A lot of people "tuck there chin" by bending there spine and neck forward and hunching there shoulders upwards towards their chin. When you do that you put your body in a unnatural position. You are less balanced which effects your ability to punch and you have lost the structural integrity behind your body. If you have good posture, chest up ,chin inwards and not downwards your chin will still be protected by your shoulders without even removing them from the socket. This prevents shoulder injuries and increases power. Also when your spine is aligned with your neck and head your ability to get hit and not be thrown of balance increases tenfold. Look at fighters like Rigondeaux or Hopkins who have extremely good posture and how there defense is benefitted by it.