The American Federation of State, County and Municipal Employees announced that the US Conference of Catholic Bishops released a statement, Respecting the Just Rights of Workers: Guidance and Options for Catholic Health Care and Unions, which provides important guidelines on unionization in Catholic hospitals. Many hospitals, including Catholic hospitals, have been hostile to unions. In contrast, at least one previous Pope (unfortunately, I don't recall who right now) made statements favorable to unions.
While I disagree with a number of Labor's stances, it is a fundamental human right for workers to be able to organize, if they choose. It is deeply wrong for any employers to interfere with their choices, and to the extent that religious hospitals were doing so, that would be doubly shameful.