I don't know a lot about veganism, but the idea that you don't eat eggs suprises me. I mean, the chickens lay them anyway. Is there some objection to how chickens are raised and forced to lay? What about free-range chickens? which I bring up since Hil mentioned how many eggs would have been in a flourless chocolate torte.
For the eggs, it's largely about the treatment of the chickens. (Basically, treated horribly, even free-range ones, and generally killed when they're too old to lay eggs anymore.) I've seen vegan message boards get into epic wars over whether someone who keeps his or her own chickens as pets, takes very good care of them, cares about them, lets them have a huge yard to run around in, and eats their eggs, can still call him or herself vegan. The general consensus on that one seems to be that the chicken-keeper is not unethical, but does not fit into the definition of "vegan" anymore, either. At one point, I saw this get into an argument about whether it was vegan to take the eggs from the pet chickens and feed them to your pet dogs. (A lot of the definition protection comes from people getting frustrated with friends or restaurants giving them some food that contains eggs or dairy and justifying it with "My sister/cousin/whatever is vegan and she'll eat that!" So a lot of vegans get kind of antsy when they see the label "vegan" being applied to people who eat things that they won't eat.)
As for the honey, it's exploitation of the bees. (I do eat honey, and know several other people who call themselves vegan who do, too.) The basic argument would be that making honey is something that the bees do naturally, for the survival of their own hive and species, and it's wrong and selfish for humans to take advantage of their labor for our own wants.
Some vegans -- who usually term themselves animal liberationists or animal abolitionists -- even say that keeping pets is wrong, because you're subjecting the pet to your own schedule and wants. (If you actually follow these arguments all the way through, they usually end up at saying that domesticating those animals was wrong in the first place, and what we should be doing is spaying and neutering as many pets as possible, so that there won't be more animals born into that life. I disagree, and this is far enough from my own position that I haven't really read into it fully and can't totally explain the arguments, and I'm sure that someone who knows more about it would probably point out a ton of errors in this paragraph.)
(I really have no stake in the argument over the eggs from the pet chickens. I won't eat eggs from farmed chickens, and I've never known anyone who kept pet chickens, so it hasn't come up. Also, I got food poisoning from an egg sandwich several years ago, and the smell of cooked eggs has made me feel sick since then, so I probably wouldn't even get to the point of making that decision anyway.)