If I hit a unit with 4 might with a Wages of Pain on the Forbidden Waste battlefield and then move in with a unit, can my opponent react with a Hidden Zhonyas to save their card at the battlefield or does it die automatically from the move trigger?

Yes, your opponent can react with a hidden Zhonya's — but only if they do it before Wages of Pain resolves.

Here's the full sequence:

1. The 4 Might unit at Forbidding Waste has -2 Might from the battlefield (defending alone), so its effective Might is 2.
2. Wages of Pain goes on the chain. Your opponent gets a priority window here (Closed State, Reaction timing). They can reveal hidden Zhonya's now — it enters as a gear permanent with its replacement effect active.
3. Wages resolves, dealing 3 damage. The unit now has 3 damage on 2 Might = lethal.
4. Cleanup occurs. During cleanup (Rule 320), "Legal Items cannot be executed" — meaning the opponent cannot play Zhonya's from hidden once cleanup has started. The unit would die here.
5. If Zhonya's was already revealed in step 2, its replacement effect fires in cleanup: "kill this instead. Heal that unit, exhaust it, and recall it" — saving the unit.

If the opponent passes priority and lets Wages resolve without revealing Zhonya's, the unit will die during the subsequent cleanup with no window to save it. There is no "move trigger" that kills it — the lethal damage from Wages is what kills it during cleanup, and the move action happens after that cleanup is complete (per FAQ #11041).