For a more elementary proof, you might want to show that if $f$ is Riemann-integrable on $[a,b]$ with $m\leq f(x)\leq M$ and $\phi:[m,M]\rightarrow\Bbb R$ is a continuous function, then $\phi\circ f$ is Riemann-integrable on $[a,b]$.
In particular if $\phi=(x\mapsto x^2)$ and $f$ is Riemann-integrable on $[a,b]$, we will get that $f^2$ is Riemann-integrable on $[a,b]$, and that gets you where you want to go.
Since this is a pretty old question and the existing answers are hints, I'll be pretty terse in the outline of this proof and let the interested fill in the details.
- You want to use the fact that a continuous function on a closed interval is...
- Recall that a function is Riemann-integrable on an interval $[a,b]$ if and only if
for all $\epsilon>0$ there is a partition $P$ of $[a,b]$ such that $U(f,P)-L(f,P)<\epsilon$.
- You'll want to split up the underlying partition of the interval $[a,b]$ you're using into two sets. One set will use the fact that $\phi$ is ..., and the other set will use the fact that $\phi$ is bounded and $f$ is Riemann-integrable.