Does each person have a particular True Nature, according to Buddhism?
To the best of my knowledge Essentialism is not a part of any school of Buddhism. Essentialism deals with metaphysical essentialism, and is essentially on the opposite end of the spectrum from emptiness/sunyata, which might be refereed to by some as Metaphysical anti-realism. However I have not met many Buddhists that want to jump into that classification. Taoists love it though. The Buddhists will assert that there is an Ultimate truth, and so can never truely be metaphysical anti-realists. At least not in the most extreme case of the view. They do show some grey areas from time to time.  

A quick dirty and brief summery of metaphysical anti-realism is that there is no "Truth"
but some views are more true than others. It is found in various places within eastern philosophy.

Also Buddhism is not nihilistic, never do they throw up their hands and say, nothing matters do whatever you want. The only term with nihilism in it that might come close to paralleling with the Buddhist worldview that I can think of is Mereological nihilism.