Can I still use an action if my only unit in a battlefield I'm attacking is pushed out by Overzealous Fan?

Yes, you can still use an action.

According to RiftJudge FAQ #8798, the combat showdown does not end immediately when a unit is removed from the battlefield by Overzealous Fan. The showdown only concludes when both players pass with focus on an empty chain.

If your unit is moved to your base by the Fan, you still have the opportunity to play an action (such as Ride the Wind to return the unit to the battlefield, or any other valid action) before the showdown concludes, provided you have the focus to do so.