After a non-combat showdown, do units get healed?

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

According to the RiftJudge FAQ (ID 3513 and 7792):

• Ruling: Units do not heal during or after non-combat showdowns. Healing only occurs after combat showdowns and at the end of the turn.
• Nuance: A showdown only becomes a "combat" when a battlefield is contested and occupied by units controlled by different players. If the showdown does not meet these criteria, it is non-combat, and no healing is triggered.

As stated in Rule 405.1, healing is the act of clearing damage from units, and this only occurs at specific points in the turn or when instructed by game effects. Combat cleanup (which includes healing all units) is specifically tied to the resolution of a combat showdown.