The title seems like a simple statement, but my lack of skill with absolute values prevents me from being too creative with the intuition. I can't seem to find an answer to this online, so I assume this result follows from a few elementary absolute value arithmetic laws.
I can reason intuitively that the sign of the quotient can never be negative because of the square and the absolute value. I have trouble proving it mathematically.
Let $x \in \mathbb{R}$. By definition, $|x|=\begin{cases} x, & \text{if }x \geqslant 0\\ -x, & \text{if }x<0 \end{cases}$
Let $x\neq0$.
If $x>0, \frac{x^2}{|x|}=\frac{x^2}{x}=x=|x|$;
If $x<0,\frac{x^2}{|x|}=\frac{x^2}{-x}=-x=|x|$
Either way, it's true: $\forall x\neq0, \frac{x^2}{|x|}=|x|$.