It's astonishing to me that Peterson and others whine so incessantly about lacking the right to be jaw-droppingly rude to transgender people. It's beyond me why he feels it's important to refuse to use appropriate pronouns for people who have transitioned gender. Oh, I realize he personally loathes the idea of transgender people; he explains why quite loudly to whomever will listen. But trans people are people whether he approves of them or not. If he can't treat trans people with ordinary civility, then complaining about free speech seems a bit more than rich.
I'm thinking of an analogous situation with the honorific "Ms" that conservative people once really hated using. You used to hear people screeching about free speech when their employers told them they could not use Miss or Mrs if a woman chose to use Ms. Oh, the howls of outrage!
But just like with trans people, it boils down to basic civility. Call people by their preferred names and titles. You don't have to agree with why the titles are used, you just have to behave with a minimum of respect, out of an understanding that people will always have ideas that differ from yours.
Peterson and people like him (such as religious people who claim their rights are infringed if they must call a transgender woman "she"), are drawing a line in the sand. They're insisting on a personal right to be rude and alienate people at work or on certain private communication forums.
It's not about free speech. It's about a non-existent "right" to treat people terribly. To create a public spectacle over a person who is simply trying to live their life in peace.
Peterson seems unable to cope with the fact that trans people exist and deserve to be left alone. I suspect he will continue to complain he gets pushback when he behaves like a jerk.