Hi Daniel,
The red/white/gold sunwheel badge is likely to date to 1942 at the earliest (not sure exactly when it was introduced but it definitely existed in 19421). It can't have existed before June 1941 because that was when the Deutsche Jugend Ungarn was formed.
The badge probably came into existence after the 1st of April 1942 when the organisation achieved official recognition (albeit with some restrictions2). Does anyone have a more exact date of introduction for the badge?
1 "The badge is similar to the HJ badge, diamond-shaped with a golden sunwheel" - Die Stimme der Minderheiten, March-April 1942 p23
2Der Volksbund der Deutschen in Ungarn 1938 - 1944 unter Horthy und Hitler, Norbert Spannenberger
3Nation und Staat; deutsche Zeitschrift fur das europaische minoritatenproblem, Volume 17 p271.