You can apply when you finish over 90 credits, which is around the end of junior year, but if you get accepted, you have to complete your bachelor's degree. All the schools I have seen require it (I have only researched accredited schools though). I think if you decide not to finish your bachelor's degree, your acceptance gets revoked (I hope Stories can chime in since he/she knows a lot on these forums). But the bachelor's degree applies to any grad/professional school in the U.S. (I am not sure if you are from U.S. or international).
Also there are combined degrees of bachelor's with advanced degrees such as med and pharm but have not seen that scenario for MPH programs. This is the only case where you are entered into their school with a bachelor's degree.
A lot of schools also require a couple of years of post-baccalaurette experience (but I am guessing they are lenient if you do healthcare for a couple of years while in school and have shown that you have demonstrated interest in the field).
So you most definitely need that bachelor's degree.