A practical reason for squaring the L2 (that is not specific to ridge regression) is that "squaring" the L2 consists of not bothering to take the square root in the first place. And since $x^2$ is strictly increasing, $||f(\textbf{x})||_2$ and $||f(\textbf{x})||_2^2$ will be optimal at the same point, so if the L2 is the optimization target (as opposed to a regularization penalty or something), it's a free speed gain. Squaring the L1 also doesn't change the optimal point, but it takes more time, so there's no reason to do it.
(If it is being used as a regularization penalty, we might still favor the faster option unless we have a specific reason to want the L2 exactly instead of just something (nonlinearly) proportional to it.)