For example, does posting something about anti-semitism reflect poorly on a candidate? How about a candidate that makes strong statements about the correlation between race and IQ?
1) If the candidate has to work on a team with others, then he has to get along with the team - or at the very least, the chemistry shouldn't be bad enough to affect performance.
2)I think it's perfectly acceptable to reject someone because you don't consider him to be morally upstanding. The second statement about correlating race and IQ is a bit ambiguous and subjective, but if, say the candidate was philosophically in favor of lynching and ethnic genocide, then I wouldn't care how good a hacker he is, I wouldn't feel right hiring him.
Posting race\IQ issue wouldn't, in my opinion, reflect poorly on the candidate even though I don't necessarily agree with it. On the other hand, posting anti-semitic remarks probably would reflect poorly, depending on how they were phrased and the context (is it just pure hate or mere opposition to certain political stances...)
And frankly, the reason and manner in which someone left their last job is a piss-poor reason for not hiring someone. What if there was sexual harassment? Stuff like that you can't know from a google search or even an interview.
That said, the real peril is that of mistaken identity. I almost missed out on a (mostly) enjoyable 2+ year relationship because my Google search revealed that the woman I was chatting with was an activist for various evangelical Christian causes. Fortunately, before breaking contact with her, I asked her about it, and soon realized that Google was leading me astray.
Posting things on the public Internet that reflects poorly on the candidate shows a judgment issue that is a valid reason for rejecting a candidate.
In the case I'm thinking of, a cursory Google search on a candidate I would otherwise have hired showed that he had at least one job that he had quit without notice; this job was not listed on his resume. In addition, this first-person blog stated that he had shown up to work so hungover he could barely see, on more than one occasion.
Should I have hired this person, given the red flags I found via Google and that there were other equally-qualified candidates who did not have those red flags?