I never have seen any characters that are in the non-binary gender spectrum. This is probably because gender is a social construct made by humans, and animals pretty much only care about the sex of the individual (male, female, intersex probably). Plus, not everyone knows about non-binary genders. This doesn't mean you can make a genderqueer dog, a character creator has the freedom to do anything with their character as long as it doesn't become a "Mary Sue".
The only non-binary gender I have seen on a character in-game was genderless, and that's often used for mythical-related characters.