Do units heal after a non-combat showdown and what is a non-combat showdown?

No — units do not heal after a non-combat showdown.

A non-combat showdown is a showdown opened at a Contested battlefield where units controlled by different players are not present (Rule 344.2) — e.g., you move a unit into a battlefield you don't control and there are no enemy units there. If units of opposing players are present, the showdown opens as a combat showdown instead (Rule 344.1).

Healing only happens in two places: during Combat Cleanup (which occurs only after a combat showdown, Rule 461) and at the end of each player's turn. Since a non-combat showdown has no Combat Cleanup, it heals nothing — even units that were at Base or other battlefields stay damaged until end of turn (FAQ #11781, #9726).