if my opponent has Brynhir Thundersong and wants to trigger hidden temporal breach to me playing a unit to base, can they do that?

No. A hidden Temporal Breach can only target a unit at the battlefield where it was hidden — it can't reach a unit at your Base.

Temporal Breach's "banish a unit" looks unrestricted in print, but when played from facedown the Hidden targeting rule forces the target to be chosen from that battlefield, and the "plays it to the same location" clause is just a linked instruction that follows the unit (CR 811.1.d.2, FAQ #12330). So the play from hidden is illegal against a Base unit. If the unit were at the battlefield where the Breach was hidden, it would work.

Worth flagging two things: Brynhir's "opponents can't play cards" only blocks her controller's opponents, so it doesn't stop your opponent from using their own Breach. And if your opponent's Brynhir trigger has already resolved this turn, you couldn't legally play a unit to Base at all — the only way that play happens is if it was already on the chain before her trigger resolved (FAQ #229, #11777).