Can you play Rengar to an uncontrolled battlefield on your opponent's turn if they try to take control of it?

No, you cannot play Rengar, Pouncing to an uncontrolled battlefield in this scenario.

According to the card text of Rengar, Pouncing, he can only be played to your base, a battlefield you control, or a battlefield you are attacking.

When your opponent moves a unit to an uncontrolled battlefield, they are contesting it. Because you do not control that battlefield and you are not the player who initiated the move (and thus are not designated as the attacker), you do not meet the requirements to play Rengar to that location.

Key Rules and Rulings:

• Attacker Status: Per RiftJudge FAQ #3963 and #427, simply contesting a battlefield does not grant you "attacker" status. You only gain the "attacker" designation when combat officially begins, which occurs after the move trigger resolves and the game returns to a neutral open state.
• Play Restrictions: Rule 739.3.a (referenced in FAQ #8779) restricts where units can be played. Rengar’s ability provides an exception, but it is strictly limited to battlefields you are currently attacking. Since you are not the attacker in this situation, you cannot use that exception to play him there.

In short, because you are not the attacker, you cannot play Rengar to that battlefield. You could, however, play him to your base or a battlefield you already control, provided you have the necessary energy.