Q: Do you have chicken for lunch?
If I want to answer this question I can say
no, I don't
I have not
You can use this reply if you are accused of having eaten the chicken.
"There was chicken in the kitchen that I wanted for supper. It's gone, I bet you have eaten it". "I have not". Here we don't have a question, but a statement was made, and another statement to counter it. If a question is asked, then the reply is "No, I didn't".
You should say, "No, I do not" or "No, I don't".
You are implying: "No, I do not [have chicken for lunch]".
This does not make any sense:
"I have not chicken for lunch"
As a general rule of thumb, to answer a question, move the subject before the predicate. Then you can remove everything after the verb and change "you" to "I". For example, "Do you have chicken for lunch?" becomes "You do have chicken for lunch" and then just "I do/don't."