Sounds like silhouette calculation. For each face of the cube you can calculate whether it is facing toward the camera or away from the camera. Then for each edge of the cube you can calculate if it connects a face that's pointing away and a face that's pointing toward the camera. If so, then it forms part of the silhouette.
Tricky to implement though, because you need to deal with floating point precision issues, and once you've figured out the silhouette edges, working out the correct ordering you want might be fiddly.
"Also, if I subdivide the cube into 8, would the same relative points for the smaller cubes also form an outline for each smaller cubes?" - With an orthographic view, I think yes. With a projection view, I think no.