Thanks for all your help! @Svetoslav Tsolov had it very close, but I was still getting an error, until I figured out the closing parenthesis was in the wrong place. Here's the final query that works:
SELECT dbo.AdminID.CountryID, dbo.AdminID.CountryName, dbo.AdminID.RegionID,
dbo.AdminID.[Region name], dbo.AdminID.DistrictID, dbo.AdminID.DistrictName,
dbo.AdminID.ADMIN3_ID, dbo.AdminID.ADMIN3,
(CASE WHEN dbo.EU_Admin3.EUID IS NULL THEN dbo.EU_Admin2.EUID ELSE dbo.EU_Admin3.EUID END) AS EUID
FROM dbo.AdminID
LEFT OUTER JOIN dbo.EU_Admin2
ON dbo.AdminID.DistrictID = dbo.EU_Admin2.DistrictID
LEFT OUTER JOIN dbo.EU_Admin3
ON dbo.AdminID.ADMIN3_ID = dbo.EU_Admin3.ADMIN3_ID