Yes, any student attending a catholic college can receive financial aid, and most students do. Financial aid can come from a variety of sources, one of the most common being the federal/state government. Students who fill out FAFSA and are qualified to receive aid from the government can apply it to tuition at a catholic college (so long as it is accredited). Catholic colleges will accept financial aid given to a student from the federal or state government regardless of the student's religious status.
Additionally, any scholarships won or awarded by a private or external source can be used at a catholic college, so long as the scholarship requirements do not say otherwise. This means most drawing scholarships, state scholarships, writing scholarships, or other scholarships won from a business, organization, church, or other source can be applied to the tuition at a catholic college. However, each school will have their own requirements regarding financial aid and payment, so be sure to check with the financial aid department for each scholarship you plan to use.
Catholic colleges also offer scholarships on a needs-basis. While students who are not catholic may not qualify for every institutional scholarship, those that are need-based typically do not require any religious statements or identity in order to be given to a student. Students may also be able to receive scholarships for academic excellence from the college even if they are not catholic. That said, each college will have different rules and requirements for their scholarships and financial aid, so check with the financial aid department of the schools you are interested in to get a complete understanding of what is necessary to receive financial aid.