This could possibly be very specific to each university, but I was wondering if there is a rule prohibiting researchers who are working within a university from conducting independent research — perhaps external funding.
This isn't a question that can be answered in general. It varies by place and by university. Some universities, like commercial organizations, will want to claim that all of your IP developed as an employee belongs to them. Others will pose no obstacle. Your best answer will come from the university you are interested in. There may be a legal affairs office that can answer such questions.
For an individual, having the university claim ownership can, possibly, have an advantage as they may support commercialization of your ideas and share the profits with you. But you can't assume that. So, it is best to ask before you begin so that you know of any limitations.