You can find the equations of the edges by intersecting the plane with the 8 faces of the cube. To do that you simply solve the system of two equations given by $x+y+z=0$ and one of
$x= \pm\frac{1}{2} \, \; \, y= \pm\frac{1}{2} \,;\, z= \pm\frac{1}{2} \,.$
keep in mind that some of the lines you'll get will not actually intersect a square face of the cube, it will pass outside the cube. Then, those are not edges of your polygon.
To find the vertices of your polygon, you solve the system given by $x+y+z=0$ and a pair of distinct variables equations from:
$x= \pm\frac{1}{2} \, \; \, y= \pm\frac{1}{2} \,;\, z= \pm\frac{1}{2} \,.$
This is easy, pick two of $(x,y,z)$ and just them $\pm \frac{1}{2}$. If the third variable is in $(-\frac{1}{2}, \frac{1}{2})$ that's a vertex, if it is outside that interval ignore it.