"To put the case for X" is a legal metaphor. When a barrister "puts the case for the defence", for instance, he puts forward arguments to explain why the prosecution's evidence is not sufficient to convict the defendant.
In your example, "put the case for essentialism" means "explain why essentialism is philosophically valid".
It's a little more difficult to paraphrase "in a negative form", since the meaning may depend on the context. It might imply (for example) that the case for essentialism rests not upon its own positive aspects, but upon a belief that it is better than any of the alternatives.
All the best,
Student or Learner