I think that another set of pictures, which show how you go about turning the cubes are in order.
I'm most impressed by your square rim with the droop. If I tried that I'm sure a snag would chip the corners.
As to the square rim of the bowl, I find that if I turn outwards from the rim of the bowl to the edge of the square taking gentle cuts, I don't chip the corners off. However, as the 'wings' become thinner and thinner, all you see is a 'ghost' so you need to be very careful of keeping fingers out of the way of the revolving corners.
As to turning the cubes, I've attached a few pics and a sketch that might help understand how they are turned..
When one side of the blank cube has been hollowed out as shown in the diagram, a stepped plug has to be turned and fitted (five of which are needed), to support the two inner cubes so that as the turning progresses, the inner cubes don't break free. A high degree of accuracy is required or the two inner cubes will be turned and drilled off-centre. If care is taken, when the turning is completed and all of the plugs are removed, each corner of the two inner cubes will be 'hanging on by a thread' and can be gently pulled free. That leaves the inner eight corners of the large cube with little internal 'pyramids' for want of a better word, which can be cleaned up with a Dremel drill and burr.
No fancy tools or equipment is called for. Two small 'tables' which fit in the toolpost of the lathe need to be made - one to support the cube while accurately sanding it square on all six sides to the correct dimensions - the other to support the turning tools held horizontally on the table while hollowing. A wooden chuck needs to be made, and two old chisels ground to shape for the two turning tools. One chisel is a 'hook tool' to undercut the cubes internally - the other a 'square end' tool to turn the larger hole and to clean the surface of the inner two cubes. The sanding disc is attached to the wooden plywood disc with Velcro and the wooden disc held on a chuck with internal dovetail jaws. The wooden chuck for the cube blank is held on external dovetail jaws.
The technique featured in Woodturning Magazine (issue 127) many years ago by Dave Springett, who is quite an expert on such things, and has written a book entitled 'Woodturning Wizardry' which can often be found quite cheaply second-hand. All rather too advanced for my somewhat limited skills!
Woodturning Wizardry by David Springett - AbeBooks
I hope the pics and sketch give some idea of what's involved. Like so many things in life, the only time I've ever felt like doing it twice is just before I've done it once! I guess I've made a dozen or so over the years as conversation pieces for friends. It takes me about a day to make one.
David.