Translation:He says he does not eat any kind of meat.
Not quite. You are correct that some FORMS of alcuno are used in +ve and -ve sentences, and maybe that is what you meant, but only the plural (alcune/i) can mean 'some'.
To be clear, I was attempting to point out to gmj1892 the conditions under which alcuno translates to some or any. I haven't figured out how to get the reply right under the post I to which I was referring.
I agree xyphax. I believe the answer is simply "He says that he does not eat any meat."
I feel the "correct" answer goes too far in including the words "kind of". I don't see italian words that translate to 'kind' or 'of'. Alcuno/a is simply the adjective 'any'. Mutttley71 provided a good alternative for 'kind of' = tipo di
"Carne" is uncountable and therefore "alcuno/a" cannot be used with it. "Alcuno/a" is used to point at a set of a whole. "Carne" per se cannot be divided. Same goes for "gente" or "latte" ("*non conosce alcuna gente" is wrong). It would be different with the plural: "non mangia alcune carni(=tipi di carne)".
This is unfortunately another example that shows that Duolingo does not use native speakers or qualified native speakers.
Thanks P-Fogg. I've read Muttley77's replies which seem to focus on the fact that "alcuna carne" is unidiomatic. I am in no position to discuss this, but I was wondering why "any kind of meat" was accepted while "meat of any kind" was not. For me both versions are basically the same.
I actually have a problem with this sentence: "alcuna carne" does not sound right. I can't really say why. I would rather say "nessun tipo di carne". Not sure if it is because "carne is uncountable". "Alcuno" carries the meaning of "a part of something" and seems odd to me in this sentence.
I believe that the problem is in the difference between the ways negative conditions are formed. Your proposal (no) describes a negation of the food (a noun - meat) and their negative (not) describes the opposite (negation) of the action (verb - eat). Duo is paying attention to the kind of word in the sentence - "non/not" for verbs and "no" for nouns and yes/no questions.