My opponent has a 4 Might unit defending at Forbidding Waste, with a hidden Edge of Night also at that battlefield. I play Wages of Pain targeting his defending unit. Then, I push a Recruit to Forbidding Waste. Can my opponent react and equip Edge of Night to his unit before it dies? If so, what exactly is he reacting to? Since Forbidding Waste itself isn't a trigger, I'm not sure what would allow him to do so.

Yes — he can, and what he's reacting to is Wages of Pain itself, not Forbidding Waste. Your spell sitting on the chain puts the game into a Closed State, which is his Reaction window; the hidden Edge of Night gained Reaction the turn after it was hidden (Rule 811.6), so it's legal to play there for [0].

Two things to be clear on. The attach is not the [C] Equip ability — that's an activated ability with Main Phase/Open State timing and can't be used here — it's Edge of Night's own play effect ("When you play this from face down, attach it to a unit you control (here)"), which is a triggered ability placed on the chain after the gear enters, above your Wages of Pain, so it resolves first (Rule 340.1). It must target a unit he controls at that battlefield, so the unit has to still be sitting there when it resolves.

On the math: 4 base Might minus the -2 for defending alone is a 2-Might unit, so your 3 damage from Wages is lethal — attaching Edge of Night's +2 puts it at 4, and it survives the 3 with damage marked (damage doesn't reduce Might; it only dies if marked damage ≥ current Might). Note your Recruit arriving doesn't turn off Forbidding Waste — "alone" only checks the defender's own units, so the -2 stays.