Can I use Jax effect to re-equip Brutalized to a unit again?

Yes, you can use the legend ability of Grandmaster at Arms (Jax) to re-equip Brutalizer to a unit.

According to the FAQ regarding Brutalizer and Jax:

• Sequence: When you use Jax's ability to re-equip Brutalizer (moving it from one unit to another, or re-attaching it), the equipment detaches from its current unit, causing the unit to lose all bonuses (the base +1 Might and the conditional +2 Might). Once it re-attaches to the new unit, it will provide those bonuses again.
• Timing: You must activate this ability during your turn when the game state is Open (the chain is empty and no showdown is occurring), as Jax's ability does not have the Action or Reaction keyword.

Important Note: When Brutalizer detaches, it loses its "this turn" bonus because that effect is tied to the state of being attached. When it re-attaches, it will provide its base +1 Might and the +2 Might bonus again, totaling +3 Might for the unit it is now attached to.