No, it is not.
https://www.forbes.com/sites/tomspiggle/2018/09/28/free-speech-work-rights/#7aace24138c8
After reading through this, it seems to back up my assertion that although "most" forms of speech are protected from government interference, it doesn't mean they are protected from employer consequences.
The story you referenced says:
From a strictly technical standpoint, the First Amendment bars Congress from enacting a law that would restrict an individual’s right to say or express themselves how they choose. Over time, that’s been expanded to include many actions by state and federal government. But it’s not absolute. For one thing,
it only protects you from government actions.