Answer
False
Work Step by Step
It is not necessary that if $A$ is a square matrix with $A^2=A$, then $A$ must be the identity matrix's zero matrix.
Let's take an example.
If we have $A=\begin{bmatrix}
1 & 0& 0\\0&0 &0\\0&0&0
\end{bmatrix}$ then $A^2=AA=\begin{bmatrix}
1 & 0& 0\\0&0 &0\\0&0&0
\end{bmatrix}\begin{bmatrix}
1 & 0& 0\\0&0 &0\\0&0&0
\end{bmatrix}=\begin{bmatrix}
1 & 0& 0\\0&0 &0\\0&0&0
\end{bmatrix}$
