In English, short responses to negative questions pattern differently from full responses. Long responses show more variety. This can be illustrated with a few examples.
If A asks, "Do you have any brothers or sisters?" B can answer in two different ways. One possible response is "No, I don't have any brothers or sisters" and another is "No, I have no brothers or sisters." However, with a short response of two words that contains no verb, the only possible response is "No, none." Notice that here any cannot be used. The only possible response is the one that contains have in the full response and not don't have.
Another example is the question "Have you heard anything?" This can be answered with "No, I haven't heard anything" or "No, I've heard nothing." In a short response of two words with no verb, however, the only possible response is "No, nothing." Notice that here anything cannot be used.
One rule of English is that double negatives must not be used. Double negatives were considered grammatical at an earlier stage of the English language but not today. For this reason, students are taught to say "I have no health problems" or "I don't have any health problems." The verb don't have must not be used with the word no. In short responses, however, the double negative is grammatical. This allows constructions such as No, nothing and No, none.