Assume that the equiangular hexagon has the side-lengths $a_{1}, a_{2}, \ldots, a_{6}$. Since all angles of the hexagon are $120^{\circ}$, extending its sides we get an equilateral triangle.
It is clear that
$$a_{1}+a_{2}+a_{6}=a_{2}+a_{3}+a_{4}=a_{4}+a_{5}+a_{6}$$
that is
$$a_{1}+a_{6}=a_{3}+a_{4} \quad \text{ and } \quad a_{2}+a_{3}=a_{5}+a_{6} \tag{1}$$
which are the necessary and sufficient conditions for a hexagon with side-lengths $a_{1}, a_{2}, \ldots, a_{6}$ to be equiangular.

If $a_{1}=n+1$, $a_{2}=n+6$, $a_{3}=n+2$, $a_{4}=n+4$, $a_{5}=n+3$, $a_{6}=n+5$, then (1) is verified and we are done.