Qualifications
It depends on whether you end up working in private industry or somewhere within the government. You may be able to sneak into a government consulting position or find work with a small company with just a bachelor's degree, but most other employers will want you to have your master's in business, and you'll really need it if you want to advance in the field (source).
Employers want to know that you're able and willing to put up with many long years of school, as it makes it that much more likely you won't fall asleep during meetings. Be warned that the top few consulting firms only recruit from the top few schools (source): Harvard, Yale, Princeton, Stanford, and...that's about it. Maybe MIT and Cal Tech thrown in there as well.
The thinking is that they're selling a very high-priced product to a client who wants to feel that the product is worth it. And the only way to maintain that price umbrella is to continue to maintain an air of exclusivity. Hence the very low representation of very smart, talented, hard-working state school alum in the lush consulting hallways.