There are probably only two cases worth worrying about, the case where the circle is centered on a grid square, and the case where it's centered on a corner between 4 squares.
In the latter case, the whole circle is inscribed in a 16 x 16 square grid. In the former case, it's clear the grid needs to be 17 x 17.
How many squares does it touch? You could find out using a compass, or use algebraic geometry and the circle's defining equation, \$ x^2 + y^2 = r^2 \$ . There will be a limited number of integral solutions, which are the points where the circle traces a grid corner, and you can fill in between them to see how many squares it touches.
The above equation in integers is a diophantine equation, and its solutions are the pythagorean triples.