I'm not sure how many more ways I can put this, but "Yes" it has very clearly been established that Jason 2.0 had naturally black hair from the outset. Aside from the assorted DC Who's Who listings (Which had in the Jason 1.0 days made it clear that he was a blond who dyed his hair) which list him as having black hair, he had black hair when he first met Bruce, he had black hair in what few flashbacks to a younger age he had and so on.
Bruce could NOT have made him dye his ginger hair black to look more like Dick (as Morrison has Jason say) because he had NEVER BEEN DEPICTED WITH ANYTHING OTHER THAN BLACK HAIR from his inception.