Check with the school you plan to attend and which consulting firms recruit on campus there through the career services office. Most big name firms will recruit seniors at some point during the school year. This is a far more efficient way to get hired vs. applying as an external applicant.
Schools look for majors such as business, econ, IT, psych and related fields. They expect good to great GPAs, excellent communication and teamwork skills, and experience working in teams and groups. A relevant internship helps, too. They do value advanced degrees, so an MS would help, yes. If you don’t have work experience upon graduation, you’ll still start out as entry level and move up based on your performance.
MBA financing varies widely by firm – some have explicit policies for it, others entertain funding on a case by case basis. I imagine that the tight economy has led firms to be more judicious in who they fund.
Good luck!