My opponent plays a unit to an open battlefield, if I play a spell like Flight or Fight to push the unit back, would that still be considered a showdown?

Yes, a showdown still occurs.

According to FAQ #7200, a showdown occurs when you move a unit to a battlefield, even if there are no enemy units present. Playing a spell like Fight or Flight to move that unit back to its base does not prevent the showdown from happening; it simply changes the state of the battlefield during that showdown.

Here is the breakdown of how this interaction works:

Sequence of Events

1. Opponent moves a unit to an open battlefield: This action triggers a showdown (Rule 429.1).
2. Showdown begins: The game enters a Showdown State (Rule 339.1).
3. You play Fight or Flight:
   • Fight or Flight has the [Action] keyword, which allows it to be played during a showdown (Rule 732.1.b).
   • Because you are playing it during a showdown, it creates a chain (Rule 338.1).
4. Resolution: Once the chain resolves, the unit is moved back to its base.
5. Showdown continues: The showdown does not end just because the unit was moved. The showdown continues until all players pass focus in order (Rule 345).

Clarifications

• Regarding Fight or Flight: Fight or Flight possesses the [Action] keyword, which permits it to be played during showdowns (Rule 732.1.c.1). Additionally, it possesses the [Hidden] keyword, which allows it to be played as a [Reaction] when played from a hidden state (Rule 737.6).
• Regarding "Playing" vs. "Moving": You are correct that Rule 429.1 specifically addresses moving a unit. If a player were to "play" a unit (from their hand) to a battlefield, that action would also trigger a showdown if it results in the battlefield becoming contested (Rule 340). The outcome regarding the showdown remains the same regardless of whether the unit arrived via a move action or by being played from hand.