How can I create a language using set operation to prove a language is not regular?
It looks like you are onto the right track. However, your notation is confusing me.

I would believe the following is what you are coming at.

$$\overline{(A\cap L(b^*c^*))} \cup \{\epsilon\} = \left\{b^{n} c^{n} \mid n \in \mathbb{N}\right\}$$
where $\overline S$ is the language of words not in $S$, a.k.a. the complement of $S$. Here are the facts that you will use.


$b^*c^*$ is a regular expression.
the intersection of two regular languages is regular.
the complement of a regular language is regular.
$\{\epsilon\}$ is a regular language.
the union of two regular languages is regular.
$\{b^{n} c^{n} \mid n \in \mathbb{N}\}$ is not regular.


I will leave the rest of reasoning to you.