This is my attempt. My instructor said it was incorrect. I would like to know exactly where my mistake is.

Suppose $x^2$ is uniformly continuous on $\mathbb R$.

So, $\forall \epsilon > 0 \ \exists \delta > 0 \ \forall x,y \in \mathbb R \ s.t. \ if \ |x - y| < \delta \ \text{then} \ |f(x) - f(y)| < \epsilon$

In this case, $|x - y| < \delta \ \text{and} \ |x^2 - y^2| < \epsilon$

$|(x - y)(x + y)| < \epsilon$

$|(x - y)| |(x + y)| < \epsilon$

$\delta|x + y| < \epsilon$

Notice, $|x + y|$ is unbounded on the real line. So $x$ and $y$ can be made arbitrarily large. So there will not always be a $\delta$ that will make all values of $x$ and $y$ less than an arbitrarily small $\epsilon$.

Therefore, $x^2$ is not uniformly continuous on $\mathbb R$.