Yeah, I would definitely say that the Plague Bringer power would partially overwrite an undead's natural immunity to diseases to simply say, "Still not affected by diseases, but can spread them around." Plenty of undead that can spread diseases naturally despite their immunity.
I would argue that undead (situationally) can still be carriers of diseases without suffering from them. Obviously a skeleton won't be able to carry a blood-born disease, but a zombie might. I would look at the specifics of the disease and rule it on a case by case basis, but as a general rule of thumb I would say yes they can still carry the disease.
Ghoul fever and mummy rot are both diseases spread by undead, so I'd say that an undead can still carry a disease even if they're not detrimentally affected.
It's a reasonable way to run it. On the other hand, the Antipaladin is getting an awful lot from being undead, so I might go with the stricter reading. It says they can "still" contract diseases- it only keeps that ability, rather than granting it, and undead removes it. If I allowed it, I'd probably want it kept to minor diseases- filth fever and the low-level undead spammers, like from a ghoul or plague zombie.