Some other interesting viewpoints:

a) if you go to the website of the Association of Jesuit Colleges and Universities (AJCU) - 28 Jesuit institutions - and search for adjunct faculty, unions, etc.   Nothing comes up!

b) It appears that there is no consistent AJCU policy on this issue to date - instead, each institution makes up its own response, protocols, etc.

c) former college student of mine is at Loyola Chicago and is the VP for legal affairs, etc.  He has written letters outlining the questions and current approach.

Here is an excellent analysis from the Jesuit Post:  https://thejesuitpost.org/2016/02/tbt-labor-rights-and-jesuit-schools/

Key sections:

"..opportunity to connect more deeply with their Catholic mission and their Jesuit history of standing on the forefront of labor rights; ....definitive choices: first, to see contingent faculty as partners in mission who should be fairly compensated; ....recognizing their right to unionize."