Strictly speaking, this is not a double negative; I believe this is known as negative concord. Basically, if the sentence is in negative, every word like anything, anywhere, etc. should be negated too.
An example of a true double negative is «не могу не заметить» (i.e. “I cannot not remark”), where the double negative collapses to a positive, just like in logic.
As far as I know its not negative concord. In some languages (like in Russian or my native Hungarian) there is mode of negation which is indeed called double negation. In this case grammatical negation differs from logical negation.
In Hungarian this "У них ничего нет." sentence is "Semmijük sincs." and it means "They don't have nothing."
The sentence translated: У него ничего нет = He doesn't have anything/He has nothing.
In other words, consider ничего нет as the doesn't have anything form.
Your issue comes from the fact that you would like to write He has nothing with a single word. Unfortunately, Russians just stick to ничего нет for that.