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"Would U.S. tech workers join a union?" survey average: 67% likely
(sh.itjust.works)
This is a most excellent place for technology news and articles.
Unions are legal in all occupations. There may be restrictions on some form of collective action (i.e. the government can force strikers back to work) but organizing is never illegal.
One caveat: the legal protections of the right to unionize apply to non-supervisors. If you have people who report to you, your power to unionize is pretty limited.
There are also some specialized jobs that aren't allowed to unionize by either federal or state law: actual soldiers in the Army, certain political jobs, etc.
But for the most part, if you are employed, you're probably allowed to unionize (and protected against retaliation even in an unsuccessful union drive).