Can the word shouldn't be use in a sentence with the word no?

Yes. For example:

Shouldn't the answer be no?

Also, you just did when you asked that question.

However, generally you do not have 'shouldn't' and 'no' in the same sentence because that would be a double negative. For example:

I shouldn't have no candy left. The sentence the author was probably meaning would be 'I shouldn't have any candy left.' With two negatives in there, the sentence means 'I should have candy left' because two negatives cancel one another out.

I shouldn't have no candy. = I should have some candy.