First, I admit that the title to this brief series is double-plus-ungood. Only natural persons (God, men, angels) can be Catholic — or Jewish or Muslim — in the sense of belonging to a religion. Slapping "Catholic" (or, if you insist, "Jewish" or "Muslim") in front of something is often a dishonest way of trying to clinch an otherwise weak or non-existent argument. It allows whoever is using his or her faith as a crutch to accuse others of being in bad faith without having to make an argument or present any proof or evidence either that his or her claims are true, or that whoever is in disagreement is thereby automatically a "bad" Catholic, Jew, or Muslim.
So much for stating the obvious.
The question arose in a recent FaceBook discussion as to whether in Catholic social teaching workers have the right not to join a union, i.e., whether a state that passes a "right to work" law is in conformity with Catholic social teaching that seems to mandate unions of some form. Our response was to see what the encyclicals actually say.
What Rerum Novarum stresses above all else is workers (and everybody else) owning a meaningful amount of capital: "We have seen that this great labor question cannot be solved save by assuming as a principle that private ownership must be held sacred and inviolable. The law, therefore, should favor ownership, and its policy should be to induce as many as possible of the people to become owners." (§ 46)
The means to do this is for people to organize and work directly on the common good; the act of organization establishes the necessary equality of political status essential to being able to access the common good directly instead of indirectly through State action. Such organization must, of course, be free, or there is no true contractual relationship — a coerced contract, or an agreement between persons of unequal status is not a contract.
Unfortunately, what has happened is that the right of free association/contract in Catholic social teaching has almost universally been understood solely as the right of labor to organize for better pay, benefits, conditions, and so on — which is actually only a part of what liberty means in the economic or political order, and the organizations themselves — intended to secure equality of status through contract — are construed as the oxymoronic "status corporate bodies" based on inequality of status and distribution on the basis of need, not equality of inputs.
So, yes, one application of the principles in Rerum Novarum (and Quadragesimo Anno) is that workers should be free to join unions or not — but focusing on that is to miss the whole point of the social doctrine of Leo XIII and Pius XI, which is to effect a restructuring of the entire social order by means of the act of social justice, not retreat to a position built on inequality of status in which the great mass of people are forced to live under "a yoke little better than that of slavery itself." (Rerum Novarum, § 3)
The whole issue could be avoided, however, simply by reforming today's labor and craft unions as "ownership unions" intended to secure and protect everyone's right to own capital as well as the means to do so.