Random speech doesn't become any less random just because it is federally protected. On the contrary, it actually becomes less federally protected if it is random, because federal protection of speech requires specific time, place, and manner.
At a broader level, federally protected speech in the US isn't absolute; the Supreme Court has ruled a number of times that restrictions can be placed on organizing protests, such as you can't protest in the middle of a busy freeway at rush hour blocking traffic and use "my right to protest is federally protected" as an excuse. There's more info here [1] (PDF warning; go to page 9).
More specifically to unionizing, employees are allowed to discuss unions but employers are allowed to place some restrictions on when/where such discussions can take place. From the NLRB [2]:
> Working time is for work, so your employer may maintain and enforce non-discriminatory rules limiting solicitation and distribution, except that your employer cannot prohibit you from talking about or soliciting for a union during non-work time, such as before or after work or during break times; or from distributing union literature during non-work time, in non-work areas, such as parking lots or break rooms.
I think it's less about Management endorsement and more about notification blindness.
If you start adding wholly unrelated messages into the security warnings, people start ignoring them. We've seen the same with medical systems where doctors and nurses ignore the important pop ups because so many are just interruptions.
It's not a free for all space to put random messages into.