I believe (unfortunately without offering any evidence) that report is inaccurate.
I recollect Hannity promoting the idea of Clinton health issues on his radio talk show long before he made it a staple of his Fox show.
I suspect he doubled down on it, because he recognized poor polling numbers may impact not only the election but his own shows. The meme is used to not only bolster Trump supporters with some justification for their support but to dissuade potential listeners who may have considered other candidates into voting for a "healthy" candidate.
