In a free for all can I play an action in a showdown I'm not participating in?

Yes, in a free-for-all game (3+ players), you can play an Action in a showdown you are not personally participating in, provided you have Focus.

According to RiftJudge FAQ #6623, during an open showdown triggered by another player, all players involved in the game have the opportunity to play actions as focus passes to them. Combat and showdowns do not restrict you from playing actions that target units at any battlefield, including those not involved in the current showdown.

How it works:

1. Showdown State: When a showdown is ongoing, it creates a "Showdown State" (Rule 343.1).
2. Focus: Focus passes between players in turn order (Rule 347.4). While you are not "participating" (i.e., you don't have units at the contested battlefield), you are still a player in the game and will receive your turn to hold Focus as it passes around the table.
3. Playing Actions: When it is your turn to have Focus, you may play an Action spell (Rule 347.1). This action can target any unit or battlefield in the game, regardless of whether that battlefield is currently in a showdown or who is involved in it (FAQ #6623).
4. Chain Rules: Remember that if you use your focus to play an Action, you start a chain (Rule 342.1). The showdown will then enter a Closed State, and only Reaction cards can be added to that chain until it resolves and the state becomes "Open" again (Rule 310.4, 325, 342).

Summary: You do not need to be a participant in the combat or showdown to interact with it; you simply need to wait for Focus to pass to you in turn order. Once you have Focus, you may play any legally timed Action or Reaction spell.