The actual idiom is "to not have a ___ bone in one's body" so you can put any adjective in there. But you can't say HE DOESN'T HAVE A BAD BONE without saying IN HIS BODY. https://www.idioms.online/not-have-a-bone-in-ones-body/
Someone who's bitchy is always bitching = complaining about something, speaking of men and women. But it can also be used for women exclusively when they're arrogant, mean and unapproachable.
In this post, Suzy hasn't given us any information or context, so I have no idea what BICHY BONE means, so I had to assume it was originally saying that someone didn't have a biTchy bone in their body.