Let's step back a minute.
First Amendment covers the right to associate (meet)
.... the right of the people peaceably to assemble....
So if workers, gun owners, hobbist, skill professionals, want to band together, for a common cause, (think, AFL/CIO, AARP, NRA, ARRL, AMA) that's protected.
WHAT's shouldn't happen is two things that have.
No group should be given exclusive control. Call it right to work, but only in the union area has this been granted. There are choices, both in social, hobby, and other skilled areas. But labor unions in some states and industries have been allowed to become the soul source of labor. That should not be. It is called in all other cases anti-trust.
The other thing is that while government employees can organize, BUT THEY should not allowed negotiate compensation and benefits, because there is no profit motive to keep things in check.
And the founders of AFL and CIO were very much anti-socialism. Gompers and Meany were definitely in favor of capitalism and very much against government employee unions.