I doubt she'll get sick if the cats are healthy. About the only thing to worry about would be if the cats had worms.
I don't know why, but most dogs view the cat litter box as their own personal Whitman's Sampler.
It might help if you set the cat box up on an old table out of the dog's reach. I'm assuming, since you say she's squeezing through the hole for the cats that it's not the Anatolian!
I seem to remember someone posting that there is something you can add to your dog's food that makes cat poo taste like, well cat poo.