On my turn, if I'm attempting to conquer a battlefield, and the enemy plays "Rengar, Trophy Hunter", if I use Flash to move my unit back to base, does the enemy score a point?

Yes, the opponent can score a point in this scenario, provided they have not already scored that specific battlefield during the current turn.

The outcome depends on the final state of the battlefield after all effects have resolved.

1. Sequence of Events

1. Your Play: You initiate a move toward a battlefield to conquer it. This initiates a non-combat showdown.
2. Opponent's Response: Your opponent plays Rengar, Trophy Hunter as a [Reaction]. He enters the battlefield. This places Rengar on the chain.
3. Your Response: You play Flash as a [Reaction] in response to Rengar being played. You target your unit(s) at that battlefield and move them back to your base. Flash is placed on top of the chain (LIFO resolution).

2. Chain Resolution (LIFO)

The chain resolves in reverse order:

1. Flash resolves: Your units are moved from the battlefield back to your base.
2. Rengar, Trophy Hunter resolves: He enters the battlefield.

3. Will the opponent score?

Whether the opponent gains a point depends on the state of the battlefield after the chain resolves:

• If you have no units remaining at the battlefield: Your opponent is the only player with units present. Per Rule 461.5, when no combat is staged, the player with units remaining at the location Establishes Control.
• Conquering: Per Rule 461.5.d, establishing control results in a Conquer action if the opponent has not yet scored that specific battlefield earlier in the same turn. If they meet this condition, they will gain 1 point.
• Combat: Because your units were moved to your base by Flash, there is no longer a combat showdown between your units and Rengar. Therefore, the combat is cancelled, and the process moves directly to establishing control and potentially conquering, as described in the interaction regarding Rengar and movement spells (similar to RiftJudge FAQ #10522).

In summary: If you vacate the battlefield entirely with Flash, your opponent will be the only player left at that location. Consequently, they will establish control and successfully conquer the battlefield to gain a point, provided they haven't already scored it this turn.

Note: I could not fully verify this ruling against the retrieved rules.