If you react to the enemy's temporary trigger, do the units die?

Yes, you can react to an enemy's Temporary trigger, and yes, the original unit will still die.

According to Rule 816, Temporary is a triggered ability that behaves like any other triggered ability (it is placed on the chain). As supported by RiftJudge FAQ #9384, because it uses the chain, players have the opportunity to play [REACTION] speed effects in response.

How it works:

1. Trigger Placement: At the start of the Beginning Phase, the Temporary effect triggers for the enemy unit and is placed on the chain. This puts the game into a Closed State.
2. Reaction Window: You (and your opponent) may now play [REACTION] cards.
3. Resolution (LIFO):
   • If you play a [REACTION] (like playing a unit or a spell), it is added to the top of the chain and resolves first.
   • Once all reactions on the chain have resolved, the original Temporary trigger resolves, and the unit is killed (sent to the trash).

Key Nuance:

If you react by playing a new temporary unit (such as with Sprite Call), that new unit will survive the turn. This is because the "Temporary" trigger check only occurs once at the start of the Beginning Phase. By the time your new unit enters the battlefield, that window has already passed, meaning it won't be checked for its "Temporary" death condition until your next Beginning Phase (as detailed in RiftJudge FAQ #340).