According to this answer about quoting a question in the middle of a sentence, the punctuation in this example sentence is correct:
People are not merely asking, "why should I do this?" they're asking, "why should anyone do this?"
But not only does it look like two separate sentences to me, maybe it should be:
People are not merely asking, "why should I do this?" They're asking, "why should anyone do this?"
If it's made into two sentences, then I'm not sure the second one is complete enough, or supported enough by the context of the first sentence, to stand on it's own.
Are they both grammatical? If so, are the two examples equivelent? Do they take on different nuances?