If your unit has 6 might and 5 damage on it. Then you use Jax to move the gear to a new unit. Does the unit die? Or does it need to be damaged again?

Yes — it dies if the gear was providing Might. When Jax moves the gear away, the unit's Might drops. If that reduced Might is now equal to or less than the 5 damage already marked, the unit will be killed during the next Cleanup (Rules 323.5, 143.2.a).

Example: A unit with base Might 4 has a gear giving +2 Might, making it 6. With 5 damage marked, it's alive. Jax removes the gear → Might drops to 4. Since 5 damage ≥ 4 Might, the unit dies at the next Cleanup.

It doesn't need to be damaged again — the damage was already there. The death check is: non-zero damage ≥ current Might. Lowering the Might below the existing damage is enough.

Only exception: If the gear you moved wasn't providing Might (e.g., it gave a keyword or ability instead), the unit stays at 6 Might with 5 damage and lives.