This is an example.
He says "Darwinian and neo-Darwinian" theories of evolution are "logically incapable" of explaining consciousness and self-awareness. Why? Because teleology is not allowed and there is no reason for beings to be "self-aware" when they could just evolve the mechanisms that make them fit sans self-awareness. The best Darwinians can hope for is that consciousness is a byproduct of some other process.
Self-awareness seems to just be an organism's capacity to recognize that she exists and persists in an environment in a generalized way. It seems to me there are all sorts of situations where taking in this information, processing it, and reacting accordingly would be useful for an organism.
Of course this capacity could be evolved separately, with no sort of general self-awareness. That's presumably what Gene has in mind: something like a computer that responds to specific stimuli as it needs to. But it doesn't seem that hard to conceive of why a brain with generalized self-awareness would be far more fit than this sort of piece-by-piece evolution of stimulus and response. A self-aware brain along with abstract thinking capacity is going to be even more robust to novel risks. Why is teleology needed? A sense of self-in-the-world is surely going to get an evolving brain further along than a disjointed assortment of stimulus and response, right?
Gene Callahan's option of direction of course is not this easy to understand. It begs the question: whence the director? Unless "well it just happened" is to count, we're in a bit of trouble. Every theory is going to run into a snag at some point - perhaps a soluble snag, perhaps not. But it seems to me we'd like our snags to be as fundamental and non-complex as possible. Explaining a grand architect seems like a bigger, less fundamental snag than explaining the existence of mute matter. Notice that this is not to say that it is less true/ It's just those that appeal to such a thing should not go around scolding others for "logical incapabilities".