Solution:

A rectangle is determined by choosing two of the nine horizontal lines and two of the nine vertical lines, so $r = \binom{9}{2}^2 = 36^2 = 1296.$

A $k \times k$ square can be placed in $(9 - k)^2$ positions, so $$s = \sum_{k=1}^{8} (9 - k)^2 = 8^2 + 7^2 + \cdots + 1^2 = \frac{8 \cdot 9 \cdot 17}{6} = 204.$$

Then $\frac{s}{r} = \frac{204}{1296} = \frac{17}{108},$ which is in lowest terms, so $m + n = 17 + 108 = 125.$