In broad terms, the First Amendment prohibits public employers from retaliating against employees who engage in "protected speech".   (Connecticut has a statute, Conn. Gen. Stat. Sec. 31-51q that purports to apply the First Amendment to private employers too.)  But proving these cases remains difficult for employees.  

And even victories may later end up as

For a few years now, I’ve been describing how social media policies are moving into the mainstream. 

No longer can employers simply cover their eyes and ears to what is going on with Facebook and Twitter.

Example No. 592: West Hartford, Connecticut is considering a policy that would place certain restrictions on what school employees say