I think it's a trick question, in the way he phrases it. If A implies B, then not B implies not A. But he doesn't say that. He says, "If A implies B, does not A imply not B," for which the answer is no.
The way he phrases it is "What can we say about !A and !B," according to the article. The only time he talks about "If A implies B, does not A imply not B," is when describing close answers he got to his question. I don't think it's a trick question the way he asks it, and it also seems like the sort of thing you should be able to work out with concrete examples if you aren't sure.
God I feel like an idiot for questioning this, but is it really true not B implies not A in that situation? It seems like it depends on what you mean by "implies."
E.g., you could have A -> B, and A -> C, and B != C. Then C is not B, but implies A just as much as B might (in the very least it doesn't imply not A per se, as A might be true). It seems like there's some implicit assumptions going on.
"not B" doesn't mean "something else which is not identically B", it means "the negation of B". Also, the statement "A->B" has absolutely no bearing on the statement "B->A". Hope this helps.