If opponent has Dazzling Aurora, and I have Brynhir with a hidden Temporal Breach, can I play Breach and replay Brynhir as a reaction to stop him from playing the unit on the Aurora trigger?

Yes — but only if Brynhir is at the battlefield where Temporal Breach was hidden.

Here's the sequence: When Aurora's end-of-turn trigger goes on the chain, you play Temporal Breach from Hidden as a Reaction (it costs 0 and has Reaction while facedown). Temporal Breach banishes Brynhir, then you play her back to that same location, ignoring cost. Her "When you play me" trigger lands on the chain above Aurora's trigger and resolves first — opponents can't play cards for the rest of the turn. When Aurora then resolves, your opponent reveals and banishes the unit, but the "play it" instruction is skipped as impossible; the rest is recycled.

The one catch: a spell played from Hidden must choose its targets from among units at the battlefield where it was hidden (Rule 811.1.d.2). So Brynhir has to already be at that battlefield — a hidden Breach can't reach her in your Base. If she's there, the play works exactly like the Thrill of the Hunt + Brynhir line against Aurora.