The eagle was worn on the DJ winter cap but only for those with rank (I'll dig those rank ranges out later when I have more time). All others wore just the HJ diamond where he has that eagle.
Edit: the black winter cap with eagle was also worn by members of the HJ Führerkorps on the skiing uniform. When I have time I'll go into more detail on this but for now, yes the regulations show that both the HJ and DJ wore the eagle on the winter cap under certain circumstances.