Although I believe it's a long-settled matter that fire departments and other public organizations can regulate certain aspects of their employees' (including volunteers) speech, I'll leave it to the lawyers — real lawyers, not firehouse lawyers — to comment on the actual First Amendment implications of this story.
Because for those of us who choose to be in public service (career, volunteer or otherwise), this story really isn't about freedom of speech, it's about trust. It's about the trust that everyone in our communities has, and must have, in their local fire department to serve them without regard to race, ethnicity, gender, religious beliefs, sexual orientation, socioeconomic status, or any other personal/familial characteristic.
Firefighters are often lauded as the "most-trusted" people in our society. That kind of trust is sacred, and it comes at a price: you can't just say (or do) whatever you want, whenever you want, to whomever you want.
You can certainly choose, as an individual, if that's a trade-off you want to make. But engaging in hateful speech — and here in the United States in the 21st century a call to "lynch" anybody is, by definition, hateful — erodes the trust that all of us in the fire and emergency services rely on to do our jobs, effectively and safely, every day.