The dog one was
clearly directed at people who insist on using the pronoun which fits a trans person's biological sex despite that being ****ing ignorant and really rude. We all make mistakes, but anyone that will knowingly continue to call someone 'he' or 'she' when they wish to be known otherwise (for example people that called Luke A 'her' or Nadia 'he') is a dick, and there's no justification for that. That's obviously what that post was about...
I see nothing wrong whatsoever with the non-binary genders petition though
and a little off-topic but I'm not sure if it was posted on here recently but Facebook has announced that people will soon be able to choose a large number of non-binary terms, so if a ****ing social networking site can lead the way, I'm sure the American government can follow suit (like other international ones have)