Suppose a person who works for the government discovers that his or her employer (i.e., the government) is monitoring its employees' social media posts during NON-work hours.
Which government? Federal? State? Local? If not federal, in what state is this happening? What agency does the employee work for? Is that same agency doing the monitoring or a different agency? What exactly does "monitoring" mean? Did you mean to say that the monitoring is occurring during non-work hours or that the social media posting is occurring during non-work hours (or both)?
Given that the employer is the government in this case though, would the employer need a search warrant to do this?
No search warrant is needed for that which is exposed to the public. If the social media posting available to the general public, it's perfectly legal for ANY employer to "monitor" it.
the idea that your employer-the-government is almost kind of stalking you during your non-work hours in order to aggregate and review everything you say in public is kind of creepy.
That's a perfectly reasonable opinion that others may or may not share, but it has no bearing on the legality of this.
It seems different in kind from (e.g.) a situation where you go to a public square to speak about some issue and by happenstance you run into your boss there, who happens to hear you speak.
It's obviously different because, here, you wrote that the employer heard the employee speak "by happenstance," whereas, in the case of the social media posting, you wrote that the employer has specifically sought out the employee's social media posts. If that distinction is removed (i.e., if the employer followed the employee to the public square with the intent of listening to the employee speak, it is pretty much exactly the same thing.