Should a research statement for a faculty job have references in it?

Yes. Definitely. This gives the reader an indication of how your work fits into the larger body of research in the field.

Without references, it may appear that you are working in an area that is so obscure that nobody cares about it.

One very important function of a research statement is to demonstrate that you know what you want to do from day 1. No one wants to hire a new faculty who have only vague ideas about what to do, since everyone is supposed to be productive as soon as possible.

From that perspective, a research proposal without specifics can be a huge red flag, and evaluators may fear that you just made up something fast. On the other hand, you can give an impression that you already researched the literature, thought through details and specifics, i.e. have concrete details in your proposals with bibliography, people may have more confidence in you. Note, that majority of the people who filter and judge your CV have no clue about your field, so such superficial impressions can be important.