I would expect Shaq to be very effective defensively against Bogut. More so than Jermaine.
Shaquille O'Neal has much more size (7-1) and length than Perkins. That combined with his strength makes him a very difficult matchup for a solid but unexceptional athlete like Andrew Bogut.
The problem with Bogut is that he is he can score in so many ways, it makes it very tough for a guy like Shaq to stop him. He can hit the mid range jumper, he can put the ball on the floor, and he has a ton of moves to to combat Shaq using his size against him in the post. Unlike someone like Howard, Bogut knows how to counter a defender leaning on him in the post, and use it to his advantage. Since Shaqs defense pretty much relies on that at this point (much like Perks post defense does), he is a bad matchup.
JO on the other hand is more of a finesse defender. He moves his feet, and uses his length to bother shots. He is not as effective against guys like Howard who simply overpower him, but he has a better shot against guys like Bogut, because he is able to recover better.