Okay.
Did a little more digging.
I partially concede.
Though the J pronounciation sounds wrong to me- it could be argued that either G or J for G is correct.
The syndrome is named after the guy who discovered it. And the guy who discovered it is himself named after something else. And that something else, if you pronouned it in the English language way, would be with a J sound for the G.
(But .if you're claiming that "J is right because that how they say it in German" you're claiming the right thing for the wrong reason.)
If your last name is "Smith" you're named after your medeaval ancestor's profession -which was banging out horseshoes with a hammer.
Likewise Dr. Hans Asperger's family name came from the word "asperge" which is the ritual of sprinkling holy water (over people and the alter) in a Catholic church.
The word was introduced into most European languages from Latin ( including both English and German).
In English the word 'asperge" is indeed pronounced "asperj".
So if you Anglicized it it could be argued that yes "asperjers" would be a correct way to say it. Either way would be ok.
However im pretty sure that the Doctor himself called himself "Asperger [with the G as in Green]" because when Germans say the name of their own country in English it usually comes out like a growl- "Grrr- many" because German speakers dont do that thing of saying J for G.