I would say "a square"; for example: "Write your answers in the squares." Maybe the British have a different way of saying it though.

This kind of notebook has a grid, this grid is made up of many squares (boxes or cells).

I'm in the US, and would tend to call it graph paper. The pattern of the lines would be called a grid.

In the US this type of notebook is not common, so we don’t have a standard term. If someone referred to what your asking about as a square, cell, or box, I would understand what he meant.

Boxes