The use of "inte någon" ('not any') instead of "ingen" ('no') might be more common in speech than in writing (and you would pronounce it "inte nånn" as long as you not put emphasis on någon). Anyway, I wouldn't call it unnatural, but a normal variation in expressing lack of something :)
I don't have a pillow is an accepted answer and I have no pillow is the suggested answer as you can see above.
I don't have any pillows would be Jag har inga kuddar in Swedish.
But you're right that I don't have any pillow is unidiomatic in English. You don't really use any with concrete nouns in the singular. So I'm removing that one as an accepted answer, but that's pretty much all I can do. Please try to avoid going crazy.
I just wanted to thank Swedish duolingo for allowing "I have no pillow." or whatever one does not have. Italian duolingo does not allow "I have no... ", even though it is a direct translation, in Italian, just as it is in Swedish. Italian duolingo does however allow, "I have got no... ". Although it is now widely used in the US. It certainly not used by people who were taught proper English. Please don't take this as affront, I think you should be able to use whatever you like and duolingo accepts. How someone else wants to express themselves is none of my business. I do enjoy expressing myself, in the more accepted English of the academic world. So thank you duolingo Swedish!