Proof Assume there exists a regular $n$-sided polygon $A_{1} A_{2} A_{3} \cdots A_{n}(n \geqslant 7)$, whose vertices $A_{1}, A_{2}, \cdots, A_{n}$ are all integer points.

Take any integer point $M$ in the coordinate plane, and draw vectors $\overrightarrow{M B_{1}}, \overrightarrow{M B_{2}}, \overrightarrow{M B_{3}}, \cdots, \overrightarrow{M B_{n}}$ such that $\overrightarrow{M B_{1}}=\overrightarrow{A_{1} \vec{A}_{2}}, \overrightarrow{M B_{2}}=\overrightarrow{A_{2} A_{3}}, \overrightarrow{M B_{3}}=\overrightarrow{A_{3} A_{4}}, \cdots, \overrightarrow{M B_{n-1}}=\overrightarrow{A_{n-1} A_{n}}, \overrightarrow{M B_{n}}=\overrightarrow{A_{n} \overrightarrow{A_{1}}}$ (as shown in Figure 6-1).

Since $A_{1}, A_{2}, \cdots, A_{n}, M$ are all integer points, by vector coordinate operations, it can be known that $B_{1}, B_{2}, \cdots, B_{n}$ are all integer points.

It is also easy to see that $\left|\overrightarrow{M B_{i}}\right|=\left|\overrightarrow{M B_{i+1}}\right|\left(i=1,2, \cdots, n\right.$, define $\left.B_{n+1}=B_{1}\right)$, and the angle between $\overrightarrow{M B_{i}}$ and $\overrightarrow{M B_{i+1}}$ is $\frac{2 \pi}{n}$.

Therefore, the $n$-sided polygon $B_{1} B_{2} \cdots B_{n}$ is a regular $n$-sided polygon.
It is easy to see that the $n$-sided polygon $B_{1} B_{2} \cdots B_{n}$ is similar to the $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$, with the similarity ratio being $\frac{\left|B_{1} B_{2}\right|}{\left|A_{1} A_{2}\right|}=\frac{\left|B_{1} B_{2}\right|}{\left|M B_{1}\right|}$, and since $\angle B_{1} M B_{2}=\frac{2 \pi}{n}$, then $\frac{\left|B_{1} B_{2}\right|}{\left|M B_{1}\right|}=2 \cos \frac{\pi}{n}$.

So $B_{1} B_{2}=2 \cos \frac{\pi}{n} \cdot\left|A_{1} A_{2}\right| \leqslant 2 \cos \frac{\pi}{7}\left|A_{1} A_{2}\right|$.
This process can be repeated indefinitely, i.e., from the original regular $n$-sided polygon, a series of regular $n$-sided polygons can be obtained, each with a side length no greater than the previous regular $n$-sided polygon's side length multiplied by $2 \cos \frac{\pi}{7}$, i.e., for any $m \in \mathbf{N}_{+}$, there exists a convex $n$-sided polygon with all vertices being integer points, whose side length is no greater than $\left(2 \cos \frac{\pi}{7}\right)^{m} \cdot\left|A_{1} A_{2}\right|$, but $2 \cos \frac{\pi}{7}<1$, so when $m \rightarrow +\infty$, $\left(2 \cos \frac{\pi}{7}\right)^{m} \cdot\left|A_{1} A_{2}\right| \rightarrow 0$, while the distance between any two different integer points is at least 1, i.e., the side length of the regular $n$-sided polygon is no less than 1, leading to a contradiction.

Therefore, there does not exist a regular $n$-sided polygon $(n \geqslant 7)$ whose vertices are all integer points.