If Jax unmatched is in play, can the player controlling Jax equip a gear that is already in play as a reaction during another player's turn?

No, you cannot use the Equip ability of a Gear already in play as a reaction during your opponent's turn, even with Jax, Unmatched on the battlefield.

Here is the breakdown of why:

1. Jax, Unmatched's Effect: Jax grants the Quick-Draw keyword to your Equipment cards "everywhere." According to Rule 745.1.d, Quick-Draw is functionally short for "[Reaction]" and "When you play this, attach it to a unit you control." This allows you to play the Gear card from your hand at Reaction speed.
2. Equip vs. Quick-Draw: The Equip ability (Rule 744) is a separate activated ability printed on the Gear card. Jax's effect does not grant the Reaction keyword to the Equip ability itself; it only grants Quick-Draw to the Gear card as a whole.
3. Timing Restrictions: According to Rule 148.2, the activated ability of Gear (which includes Equip) can only be executed during your own Action Phase in a Neutral Open state. It cannot be activated during an opponent's turn or during a Showdown.

Summary: While Jax allows you to play Gear from your hand at Reaction speed, it does not change the timing restrictions for activating the Equip ability on a Gear that is already on the board. You must still follow the standard timing rules for activated abilities (Rule 148.2).