This is one of many interesting free speech questions raised in Carter v. Transport Workers Union of Am. & Southwest Airlines Co. (N.D. Tex.). Let me discuss those questions in turn.
[1.] We begin with the employer's punishment of employee speech that led to the case in the first place:
In 2017, a flock of Southwest flight attendants descended upon Washington, D.C. for the so-called "Women's March" protesting President Donald J. Trump. Those flight attendants were also Local 556 members, and one—Audrey Stone—was the union's president. During the march, they carried signage emblazoned with Southwest's logo. Carter, a pro-life Christian, objected to the flight attendants' purported representation of Local 556, Southwest, and, by extension, herself—especially given Planned Parenthood's sponsorship of the protest. To voice her objection, Carter sent Stone several private Facebook messages, some of which contained videos of aborted babies. In one of those messages, Carter said, "[t]his is what you supported during your Paid Leave with others at the Women's MARCH."
Stone filed a complaint against Carter with Southwest. After a hearing, Southwest fired Carter, saying that when Carter messaged Stone, she was "identifiable as a Southwest Airlines Employee and represented our Company in a manner that is disparaging to Southwest Flight Attendants."
Carter sued, alleging that (1) Local 556 had breached its duty of fair representation, (2) Southwest and Local 556 had retaliated against Carter for exercising her protected rights under the Railway Labor Act …, and (3) Southwest and Local 556 had discriminated against Carter for her religious beliefs and practices under Title VII. A jury found for Carter on each claim ….