Answer: 574

Solution:

Since $\gcd(1001, 429) = 143,$ the segment passes through the lattice points $(7k, 3k)$ for $k = 0, \ldots, 143$ and consists of $143$ translated copies of the segment from $(0, 0)$ to $(7, 3).$ The line is $y = \frac{3}{7}x.$ It meets the square centered at $(m, n)$ exactly when its height passes within $\frac{1}{10}$ of $n$ for some $x$ within $\frac{1}{10}$ of $m,$ that is when $\left|\frac{3m}{7} - n\right| \le \frac{1}{10} + \frac{3}{7} \cdot \frac{1}{10} = \frac{1}{7},$ or equivalently $|3m - 7n| \le 1.$

For $0 \le m \le 7$ the solutions are $(0, 0)$ and $(7, 3)$ with $3m - 7n = 0,$ and $(2, 1)$ and $(5, 2)$ with $3m - 7n = \mp 1.$ In the first two the line passes through the center, so it meets the circle as well. In the other two, equality means the line passes exactly through a corner of the square (for $(2, 1),$ the corner $(2.1, 0.9)$), while its distance to the center is $\frac{1}{\sqrt{3^2 + 7^2}} = \frac{1}{\sqrt{58}} \gt \frac{1}{10},$ so it misses the circle. Thus each copy of the segment meets $4$ squares and $2$ circles.

The $142$ interior lattice points $(7k, 3k)$ are each shared by two consecutive copies, so $m = 4 \cdot 143 - 142 = 430$ and $n = 2 \cdot 143 - 142 = 144,$ giving $m + n = 574.$