A bijection $\phi:\mathbb R \to \mathbb R$ that must be an identity

From what you've done, it's easy to see that $\phi(q) = q$ for any rational number $q$.

Now the key point is to show that $\phi$ is monotone.

In fact, if $x\in\mathbb{R}$ is non-negative, then there exists $y\in\mathbb{R}$ such that $x = y^2$. It follows that $\phi(x) = \phi(y)^2 \geq 0$. Thus we have shown that $\phi$ maps non-negative real numbers to non-negative real numbers.

Hence if $u \geq v$ are real numbers, then we have $u - v \geq 0$ and therefore $\phi(u) - \phi(v) = \phi(u - v) \geq 0$, or $\phi(u)\geq\phi(v)$.

Once this is done, it only remains to note that $\mathbb{Q}$ is dense in $\mathbb{R}$.