Square of the Absolute Value of a Characteristic Function is a Characteristic Function

Hint:

If $X$ and $Y$ are independent then: $$\phi_{X+Y}(t)=\phi_X(t)\phi_Y(t)$$

If $Y$ and $-X$ have the same distribution then: $$\phi_Y(t)=\overline{\phi_X(t)}$$

Hint. Let $X'$ be an independent copy of $X$, then $\exp(itX)$ and $\exp(-itX')$ are independent also, but, as $X$ and $X'$ have the same distribution, we have $\overline{\phi(t)} = \mathbf E[\exp(-itX')]$. Now use what you know about the expectation of independent products.