It's not used in negative sentences, as such, but rather in negation: saying that something isn't there, or there is none of something.
Я не вижу собаку (accusative case - I don't see the dog).
У меня нет собаки (genitive case - I don't have a dog/I have no dog).
В школе - нет столов (genitive case - at school there are no tables)
My understanding (which could be wrong) is that genitive is used with negation unless a specific thing is being talked about.
Since we're saying "our" book I believe that's why the accusative is used.
I think I also read here that using accusative with negation is more conversational and using genitive is more formal.
Edit: Here's a link that might (not) help (at all).
@PerplexUse - You can, but it reads as if you're seeing someone else's book, not your own.
In "Я не вижу нашу книгу", it reads as if the person is just confirming (or denying) that they see "our book" there - there may or may not be other books present.
In "Я не нашу книгу вижу", it reads as if the person is saying, "I see someone else's book here, but not ours".