I suspect that your oven isn't going to work because of the second law of thermodynamics.

Reflective surfaces absorb less heat, it's true, but they also emit less heat. The way that solar cookers that I read about work is they reflect sunlight onto a target, and this reflected light turns to heat and does the cooking. This works because the high-temperature object being used to reflect light is the sun, so the maximum possible temperature is the temperature of the sun's photosphere. You could also make a solar oven with lenses instead of mirrors, like a giant magnifying glass.

In your case, what you are talking about building is essentially a greenhouse. The multilayered series of boxes would be great for storing heat, but I have a hard time believing that they will be able to concentrate it. You certainly will not be able to concentrate heat from just having the thing sitting around. I don't know what you'll be able to do with sunlight, but I doubt it will be an effective oven.