Is "nobody just simply cares" a correct sentence?

"Correct" is a problematic term in this case because so much of our language depends on the context. I'm going to assume that you aren't going for a question about a dialect. Thus, let's talk about this sentence in terms of standard American English.

This sentence can be viewed as "correct" as it could be a response to the statement "I simply care." This sentence would then refute the claim by saying that "Nobody (which is the subject of the sentence) just 'simply cares.'"