Poison isn't disease so no, never.
So yes RAW constructs, elementals, undead, and other odd things can be afflicted with a disease through anything that says X happens and creature becomes diseased.
I might rule against RAW sometimes but it depends. Hopefully the DMG will clear this up some, but I doubt it right now it seems like every issue people have with the system they are hoping for a fix or clarification in that book and it just isn't big enough.
The designers dropped the ball on this one I think, and should have had diseased as a condition.