I have a unit on the battlefield with a hidden Zhonyas. My opponent sends his unit with more might and wins combat. I react with Zhonyas. When my unit dies, can I react with Rengar on the chain of the Zhonyas stack to put him on the same battlefield as the dead unit?

No, you cannot.

According to the rules regarding the timing of combat and replacement effects:

1. Replacement Effect: Zhonya's Hourglass is a replacement effect. As stated in Rule 369.1 and Rule 370, it does not use the chain. It intercedes during the execution of a game event (your unit dying) to replace that event with something else.
2. No Chain to React To: Because Zhonya’s effect does not use the chain, it does not provide an "opportunity to react" or a chain on which you can play Rengar.
3. Timing of Combat: Per the RiftJudge FAQ #8150, combat damage is assigned after both players pass during the showdown phase. Once both players pass, no one can play anything until combat cleanup finishes.
4. Zhonya's Triggering: If your unit is dealt lethal damage, Zhonya's applies automatically as a replacement for that death event. Since this is an automatic replacement of the death event and not a triggered ability or a spell going on the chain, there is no moment where you are in the middle of a chain resolve that allows you to play Rengar.

Summary:

• You cannot play Rengar, Pouncing in response to the Zhonya's Hourglass replacement effect because the replacement effect does not use the chain.
• Once combat damage is dealt and both players have passed priority, you cannot play Rengar until after the combat cleanup process is complete.

This is my interpretation based on the mechanics of replacement effects as defined in Rule 369 and Rule 370. No official FAQ exists for this specific chain-reaction attempt, but the rules for replacement effects and combat timing clarify that this interaction is not possible.