Answer: 888

Solution:

The line meets the axes at $(223, 0)$ and $(0, 9),$ so all qualifying squares lie inside the $223 \times 9$ rectangle, which contains $223 \cdot 9 = 2007$ unit squares. Because $\gcd(9, 223) = 1,$ the segment passes through no interior lattice point; it crosses $222$ interior vertical lines and $8$ interior horizontal lines, entering a new square at each crossing, so it passes through the interiors of $223 + 9 - 1 = 231$ squares.

The other $2007 - 231 = 1776$ squares lie entirely above or entirely below the segment. The segment's midpoint $\left(\frac{223}{2}, \frac{9}{2}\right)$ is the center of the rectangle, so rotating by $180^\circ$ about it swaps the two groups. Hence exactly half of them, $888,$ lie below the graph.