Public school guidance counselor = certification
Independent school college counselor = experience
You certainly do not need to major in education in order to become a counselor. I majored in philosophy and religion, for example. For the college counseling piece specifically, however, private schools (especially if they're college prep) are going to look for experience in college admissions. Starting off as a tour guide or intern in your college's admissions office is a great way to get your foot in the door (much less see if you like it), and then pursuing an admissions job for at least two or three years at your alma mater or a different college is a pretty standard route to becoming a college counselor in an independent school. If you're interested in the job descriptions and/or qualifications for either position, check out the job ads at these sites:
NAIS - National Association of Independent Schools (National Association of Independent Schools - for counseling jobs)
The Chronicle of Higher Education (The Chronicle of Higher Education - for admissions jobs)
Good luck!