No, it's a simple question. Do you think they have a duty to a company that they're not working for or that they can breach a contract they haven't even signed?
Yes or no.
Things like having murdered speak to the character of the individual. They still can't be fired for that if they've been employed.
No, it's not a fireable offense to not tell your employer your whole life story. That's insane if you can do that in Canada.
No contract I have ever signed has any terms at all about past conduct having to be up to their standards on what is acceptable speech or what is not acceptable speech.