A mezuzah is a container that has within it a scroll on which the Shema Yisrael
("Hear, O Israel") prayer is inscribed. Observant Jews affix it to the doors of their homes in order to fulfill the commandment to keep these words "on the doorposts of your house" (Deut. 6:9
). This commandment is not binding upon Catholics because it is part of the Mosaic Covenant. Catholics of Jewish heritage though may wish to hang a mezuzah, not out of obligation, but out of a desire to maintain their Jewish heritage.
As to the opinion of Jews as to whether a gentile should hang a mezuzah, I doubt you could find one Jewish statement on the matter that would reflect the views of all Jews everywhere or that would speak on their behalf. But here is one sample opinion, given by a rabbi, that should be taken into consideration:
In general, I don't think its a good idea to imitate the observances of religions that are not one's own. For example, many Americans seem to admire some of the religious practices of Native Americans, like putting "dream catchers" up in their homes. But, I wonder: How do today's Native Americans feel about Americans of European descent copying this practice without really understanding the culture, history, and belief behind it? Do they view it as something that makes light of their culture? It's something that I think you have to decide for yourself.
You should also be aware that the presence of a mezuzah on a house is generally considered by Jews to be a declaration that a house is the home of a Jewish family. That's a declaration that many Jews take quite seriously. To some Jews, a mezuzah on your house may be taken as a "false declaration." I would suggest that you ask your Jewish friends what they think (source, slightly edited).