Solution:

Take compass directions aligned with the grid. Let $N$, $E$, $S$, $W$ be the most northerly, easterly, southerly and westerly points on the circle. The arc from $N$ to $E$ must cross $100$ north-south grid lines and $100$ east-west grid lines. Each time it crosses a grid line it changes square (and it never crosses two grid lines at once, because it does not pass through any lattice points), so the arc $N$ to $E$ must pass through $200$ in addition to the starting square. Similarly for the other $4$ arcs. So the circle passes through a total of $800$ squares (we count the starting square in the last $200$).