Catholic colleges have very similar qualification requirements to those of non-Catholic universities. There may be some slight differences, however these are typically involving mission and values. For the most part, qualifications for faculty members are very similar between Catholic and non-Catholic universities.
Academic qualifications is one of the most important areas that are measured by colleges of all kinds. In order to be a professor or to be considered for a tenurable position, applicants must have advanced degrees in the field they desire to teach in. These go beyond bachelor's degrees, typically being Ph.D.'s (Doctor of Philosophy), Ed.D.'s (Doctor of Education), or MFA's (Master of Fine Arts). This ensures that the individual is well-informed in the subject, has thoroughly studied/researched, and been recognized for their accomplishments in their area of expertise.
Often, experience in the field is also taken into consideration as part of a professor's qualifications. Specifically in applied disciplines, like business and engineering, professors who have professional experience working in the industry are seen as more qualified. The practical application of what they teach students is more readily understood by them and easier to pass on to students looking to have similar experiences.
An area where qualifications for a Catholic college may differ from other colleges would be mission and values. Catholic colleges will prefer professors with an understanding and reverence for the Catholic faith and values. While service is a common value at many universities, it is emphasized at Catholic colleges. Professors who are committed to service, developing the whole person, respecting diversity, and adhering to the Catholic faith are often considered more qualified at a Catholic college than professors with a basic understanding and appreciation of Catholicism.
However, in general, faculty qualifications at a Catholic college are very similar to that of a non-Catholic university, especially another private University. More prestigious colleges, Catholic and non-Catholic alike, will have more advanced qualifications and expectations of their faculty. For questions about what sort of qualifications a Catholic college maintains for its faculty, ask the admissions department of the specific college(s) you are interested in.