No. As my magnifying glass example pointed out, the object can only increase in temperature until it matches the Sun's temp. Otherwise it would be heating the Sun not the other way around.
Sadly, your "proof" also makes no sense. The temperature in the object in the focus of the magnifying glass is a function of incoming flux and the ratio of concentration. Imagine an earth-size magnifying glass squeezing the sun's flux into a square inch. You can plug the resulting flux into Stefan-Boltzmann, and do the calculation yourself. I guarantee the resulting temperature is higher than the sun's surface. Moreover, every bit of radiation absorbed by the sun, even the earth's light, transfers energy and thus "warms" the sun, just a tiny bit.
Every normal physical object radiates, and in all directions, and not just "upwards". In the example, the whole flux received by the earth concentrated on that small, square-inch object would constitute a far bigger energy flux than the tiny slice of energy radiated from that object into the direction of the sun. Think about it, Ian, you'll figure it out.