[![enter image description here][1]][1]

The image above shows a blue mesh with a sphere shape, hereafter the blue sphere.  It is symmetric about the 3 axis similar to a cube.  Note the red faces have a 3 face corner not a 4 face corner. Not a UV Sphere.  The mesh will work symmetrically along x, y, z axis with a cast cuboid modifier.

In the image in your original question concentric rings of the UV Sphere are placed along the z-axis.  Speaking about the default situation.  So the there is an asymmetry regarding the z-axis.  So the [cast] modifier mathematics amplifies this asymmetry.  You will get slightly better results if you subdivide your mesh again.

You can edit the blue sphere in [edit] mode and rotate 90 on the y axis.  The asymmetry deformation will be rotated as well.

Here is the same blue sphere shaped mesh with a cast modifier.  Note again the red faces which form a 3 face corner of a cube.

[![enter image description here][2]][2]

To create a NON-UV sphere shaped mesh.  Create a cube. In [edit] mode loop cut on with 7 or so divisions. Loop Cut on all three axis XYZ.  Menu Transform ..... Transform to Sphere.  

Yes, start with cube, transform to sphere shape, cast back to cuboid.

Additional Notes
Please welcome Carlo

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**Cast modifier** basically projects the object vertices along their normals to the target shape's surface. It is the same thing you would get by using a Shrinkwrap modifier pointing to a scene object.

[![enter image description here][3]][3]
[![enter image description here][4]][4]

By enabling Vertex normals in Edit mode, you can see on what surface each vertex will be translated to.

[![enter image description here][5]][5]
[![enter image description here][6]][6]

[![enter image description here][7]][7]
[![enter image description here][8]][8]

There is no vertex pointing toward the box's corner.

[![enter image description here][9]][9]

**Conclusions:**
To better approximate a cube, the starting sphere should have enough vertex density whose vertex normals point to the cube edges and corners. So for better results you can:

 - Increase the number of vertices
 - Use a different starting topology

  [1]: https://i.sstatic.net/ZXcf6.png
  [2]: https://i.sstatic.net/m1D4E.png
  [3]: https://i.sstatic.net/Imna9.jpg
  [4]: https://i.sstatic.net/BCRiF.jpg
  [5]: https://i.sstatic.net/RibGTm.jpg
  [6]: https://i.sstatic.net/Hnll2m.jpg
  [7]: https://i.sstatic.net/8xMNBm.jpg
  [8]: https://i.sstatic.net/YCCq7m.jpg
  [9]: https://i.sstatic.net/SkyyM.jpg