The 'Verband der deutschen Berufsgenossenschaften' (German Occupational Health and Safety Agency) which is shown on the description plate certainly existed during the period and one of its areas of responsibility was indeed first aid but that plate doesn't seem to belong to the box (rust?). However, if we go with this box being one supplied by or proscribed by the association then the life rune may not necessarily be indicating an association with the Hitler Youth, I say that firstly because of the purpose of the association and secondly because as we know, the life rune was used elsewhere too (on some birth certificates and SS paraphernalia for example).
I wouldn't think that a unit below Bann level would have something like this when they had Feldscher/GD-Mädel with their own supplies and equipment. On the other hand we have HJ working at Volkswagen as apprentices for example who were organised into 'Bann 468 VW Braunschweig-Vorwerk' and would as such have come under the auspices of the factory in terms of health and safety etc so who knows? It might be a box from a location like that. Certainly a very interesting item.