Is the sentence "Don't you cry" (or don't you speak, don't you cry etc.) is correct? For me, as ELL, it looks weird to see pronoun (in this case "you") in such sentence - as an order rather than a question. I expected it to be: "don't cry" instead of "don't you cry". what is the explanation for that?
Don't you cry.
The former sentence is a normal imperative negative sentence. The latter can also be treated so as an imperative. They have the same meaning; you are asking the listener, you, not to cry. You are using the pronoun you in the second sentence to emphasize the listener, you.