Do units still heal at the end of a non-combat showdown?

No, units do not heal at the end of a non-combat showdown.

According to RiftJudge FAQ #9726, #9016, and #9220, healing in Riftbound only occurs at two specific times:

1. During a Combat Cleanup: This step only occurs after a combat showdown (a showdown contested by units controlled by different players).
2. At the end of each player's turn: This occurs during the End of Turn Cleanup.

Because a non-combat showdown—defined as a showdown that is not contested by units controlled by different players—does not meet the criteria for "combat," it does not trigger the healing process associated with a combat cleanup.