meshlabjs | javascript client based mesh processing tool | 3D Animation library

As mentioned in the other answers the problem is your mesh normals. Computing an average normal, like you are doing currently, is what you would want to do for a smooth object like a sphere.
cgal has a function for that CGAL::Polygon_mesh_processing::compute_vertex_normal
For a cube what you want is normals perpendicular to the faces
cgal has a functoin for that too CGAL::Polygon_mesh_processing::compute_face_normal

To debug the normals you can just set fragColor = vec4(norm,1); in mainmesh.frag. Here the cubes on the left have averaged (smooth) normals and on the right have face (flat) normals:

And shaded they look like this:

shading has to work for any kind of mesh (a cube or any organic mesh)

For that you can use something like per_corner_normals whitch:

Implements a simple scheme which computes corner normals as averages of normals of faces incident on the corresponding vertex which do not deviate by more than a specified dihedral angle (e.g. 20°)

And this is what it looks like with a angle of 1°, 20°, 100°: