Opponent plays Star Crossed on my Brynhildr in base and his unit in his base. I Thrill of the Hunt Brynhildr to his occupied battlefield. Does his target go to hand and does he have a window to play Eclipse before Brynhildr play ability goes off if he reacts at right time?

Here's the full ruling:

Yes, the opponent's unit still goes to hand. And yes, they have a window to play Eclipse before Brynhir's ability resolves.

Here's the sequence:

Thrill of the Hunt resolves first (LIFO). During its resolution, Brynhir is banished and then played to the opponent's occupied battlefield. Because Brynhir moved to a non-board zone (banishment) and re-entered as a new game object, the targeting relationship between Star-Crossed and Brynhir is severed (per FAQ #10990, consistent with Rule 359.3.e.4).

After Thrill finishes, Brynhir finalizes and enters the board. Then its "When you play me" trigger goes on the chain above Star-Crossed.

Priority window opens. Before Brynhir's trigger resolves, both players can play Reaction spells — this is where your opponent can play Eclipse. The chain at this point looks like:

1. [Top] Brynhir's trigger (finalized — can be reacted to)
2. [Bottom] Star-Crossed (finalized)

Then: Brynhir's trigger resolves — opponents can't play cards this turn.

Then: Star-Crossed resolves — your opponent's friendly unit (still a legal target) returns to their hand. Brynhir is unaffected.

So the opponent does have a Reaction window, but Eclipse (-4 Shield) wouldn't stop Brynhir's "opponents can't play cards" trigger — it only reduces a unit's Shield. They'd need something that counters triggered abilities (like a counter spell) instead.