if showdown ends at a battlefield, do all units on board heal?

It depends — only if it was a Combat Showdown.

If the showdown was a combat showdown (units controlled by two opposing players at the contested battlefield), then yes: Combat Cleanup heals every unit everywhere in play simultaneously, not just the ones that fought — units at Base or other battlefields heal too (FAQ 6111, Rule 466.1.a.1).

A non-combat showdown (e.g. you move onto an empty/uncontested battlefield, or only one player has units there) triggers no Combat Cleanup, so no healing happens — all units also heal at end of turn during the Expiration Step regardless (Rule 317.2.b).