If I use Stare Down and target Kha'Zix, Mutating Horror, will Stare Down use 4 might or 6 might?

It depends — on whether Kha'Zix's +2 Might has already resolved at the moment Stare Down resolves. Stare Down compares Might at resolution, not from the printed value: it moves enemy units at that battlefield with less Might than your chosen friendly unit, and units are evaluated by their current Might (Rule 710).

So if Kha'Zix attacked or defended and his trigger has resolved, he's a 6 Might unit and Stare Down checks against 6. If the trigger is still on the chain when Stare Down resolves (or he never gained the attacker/defender designation this turn), he's 4. Also worth noting: Stare Down doesn't actually target Kha'Zix — it targets your friendly unit and the battlefield, and the affected enemy units are selected by the instruction (Rule 355.10.d), so there's no targeting window to fear on his side.