Do I need to square a column if I want a neural network to try using that?
You don’t necessarily need to, according to the universal function approximation theorem.

It is easier for a neural network to learn an identity function than some other function, so if one of the inputs definitely needs to be squared your network will learn faster if you pass the input already squared
If your network is sufficiently large it should work out that squaring that input is helpful and approximate the squaring function as part of the overall learning process