The movement ends when the charger makes the attack

Charge (PHB p287):

Move and Attack: Move your speed as part of the charge and make a melee basic attack or a bull rush at the end of your move.

Obviously you do not need to move 6 squares even if your enemy is only 3 squares away. The movement ends when you attack (or of the target moved out of your movement range).
So the DM was wrong, the Barbarian could have followed.

Were the trigger "when you are attacked", you could disrupt the attack.

The ability can only prevent the attack if it places the enemy far enough

Two things can end your move:

1. You run out of speed
2. You do something after which you cannot resume moving
   • Take a Standard or a Minor action
   • Fall prone
   • Attack, in case you were moving as part of a Charge

Rules Compendium changes the definition of Charge, compared to PHB. No mention of "ending your move" any more. Charge (Rules Compendium p240):

Move: The creature moves up to its speed toward the target. Each square of movement must bring the creature closer to the target, and the creature must end the move at least 2 squares away from its starting position.

As Glazius writes, "Each square of movement during the charge is its own separate event." So each square of movement can be reacted to.

The sequence of events:

1. Barbarian moves to adjacent square
2. The enemy has two options
   • Trigger the ability, then the Barbarian can chase if he has movement left, and attack it
   • Not trigger the ability, but then the Barbarian can attack it

So it can only help if this moves the enemy beyond the Barbarian's speed.
You can think of it as the Barbarian ending his move in each square he moves into, and then resuming it if needed.