There was a book out last year called Will in the World, which while not going as far as this book, does suggest that Shakespeare was exposed to some of the Catholic "underground." This type of thinking is on the cutting edge in Shakespeare scholarship.
BTW, thanks for posting about the book. I'm a Shakespeare fanatic and will probably end up buying it.
I watched a PBS (eek!) show called The Search for Shakespeare (I think) on DVD that explored some of the evidence for this same theory. It dragged a little bit in the third and fourth hour because of repetition but it was interesting.
The first half of 'Will in the World' was especially good.
Greenblatt's speculation that Shakespeare could have met with Campion was intriguing.