When it comes to purchasing an HD set top converter box, I like to buy either directly from the manufacturer or from a reputable store in a sealed box. These converter boxes are pretty complex internally and they aren't toys easly fixable.
Buying one of these HD boxes from E-Bay in my opinion is very risky businness and in many cases the buyer ends up to loose the money.
According to the description you're giving us, it seems to me there is something wrong in the power supply circuit with the converter box justfying the reason why doesn't go on, and I sincerely wish you good luck in returning the box.
Mr.T