No, Constrict would still deal 1 HP damage. This would remain the case no matter how many negative conditions you apply to the attacking Shuckle, or positive ones you give the defending one.
Though the damage calculator would absolutely round this attack down to 0, the game is programmed to make 0 damage impossible. It instead sets the damage to 1 HP when this happens.
The only time this doesn't occur is when you use a move like False Swipe on a Pokemon with 1 HP. In that case, the process is overridden and the damage stays at 0.
There is also a bug in Gen 1, and a different one in Gen 5, that allows you to deal 0 damage in particular circumstances. However, these are clearly development oversights.