That’s an interesting idea David! Your logic for the black paint and reflective aluminum coatings are spot on, but I think you’re getting a bit mixed up with how the heat would transfer from the outer boxes to the inner boxes. I believe your design should work conceptually, but not sure how hot it will get or how quickly it will heat up. Building a prototype might be the easiest way to find out! I think it would also be interesting to test out your prototype both with and without the aluminum foil layers to see exactly how that impacts efficiency.

To simplify your design conceptually, let’s just assume that your oven design exists in a vacuum. There’s no air outside or inside your oven, and also no air in between the boxes. This eliminates the effects of convective heat transfer, which would only reduce the efficiency of your design. In this design, heat is only transferred between the boxes through radiative heat transfer which I think you recognized.

As the outer boxes heat up, they emit some of their heat in the form of black-body radiation. Black-body radiation is just a broad distribution of many wavelengths of electromagnetic radiation (light) where the intensity of each wavelength follows a specific function of the temperature of the object. For very hot objects, this radiation falls more into the visible region of the electromagnetic light spectrum while for cooler objects, this radiation falls more in the infrared region.

Black-body radiation is what you’re seeing when a hot piece of metal is glowing red or white. It’s also how thermal imaging cameras work, as the cameras detect the wavelengths of infrared light emitted from objects and use that to back-calculate their temperature.

Ideally in your design, this black-body radiation would get focused onto the inner boxes. Since this radiation is a form of light (infrared in the temperatures you would be operating at), the aluminum foil will reflect it and the black paint will absorb it. This will work even in the dark since the boxes are essentially creating their own light (just in a wavelength humans can’t see).

A few things to consider:
The number of boxes you use isn’t so important with regards to radiative heat transfer and may actually hurt your efficiency. Since you are trying to focus the energy from the sun into your oven, only two dimensions really matter:
the sizes of the outermost and innermost boxes.
The surface area of the outer box determines how much sunlight you collect, while the mass of the inner box determines the final temperature (assuming that all the energy is focused to and absorbed by the inner box). All of the intermediate boxes just increase the thermal resistance of your design, slowing heat transfer to the center. Thermal resistance is analogous to electrical resistance, where the more resistors you have in series (in this case number of layers to your oven), the lower your current flow (or heat flux) will be. In a real system with conductive and convective heat transfer, a smaller heat flow will be more quickly dissipated, giving a lower maximum oven temperature.

Further, the rate at which objects emit black-body radiation is dependent both on the material being used and the temperature of the box. This is described by the Stefan-Boltzmann law, shown below:

In this equation, q is the heat flux (the amount of energy being released per unit time and per unit surface area), ε is the emissivity, σ is a constant, and T is the temperature of the box. To heat your oven and whatever you put in it, you want to maximize q.

This sounds like a great opportunity for doing experiments.

How will you keep the metal boxes from touching each other? Why do you think you will have a VERY efficient solar oven? How does your solar oven plan compare with other solar ovens? Here's a link to some:
link 1
and a link to one from NASA for making s'mores:
smores.

Thermodynamics is about systems at equilibrium, where they are not changing. You want your oven to change temperature, so you might want to know about the kinetics of heat transfer - how fast the heat transfers, or, the rates of heat transfer.

Here's a link about the thermodynamics of heat transfer:
Thermodynamics.