Every nursing program must be approved by the BON of the state in which it is located in order for its graduates to be eligible to write the NCLEX. This has nothing to do with NLNAC or CCNE accreditation, which is an entirely different matter. Accreditation by the NLNAC or CCNE is voluntary, and no state requires that a school be accredited in order for graduates to be eligible for licensure. People are often confused about this because so many people use the terms "approved" and "accredited" interchangeably (a few state BONs even use the term "accreditation" when they're referring to their "approval," which is very confusing).
Whether or not a nursing program is NLNAC or CCNE accredited has nothing to do with licensure in any state. However, a growing number of employers will only hire graduates of accredited programs, and these employers tend to be the better employers in healthcare, inc. the VA system and the US military, some university teaching hospitals, etc. Certainly not all employers, not even most; but why close off any future employment opportunities for yourself this early in the journey?
Also, most programs of higher education in nursing will only accept graduates of NLNAC or CCNE accredited programs.
The other accreditation issue you need to think about is academic accreditation (the accreditation of the general, entire school, not just the nursing program specifically). This is particularly an issue with the proprietary (for-profit) tech/voc schools. Most (all?) of them will tell prospective students, and highlight on their websites, that they are "accredited." Most prospective students don't know to inquire any further than that, and that's what they're counting on. Most of those schools are accredited, so they're not lying, but they're accredited by organizations that only accredit proprietary tech/voc schools. They do not meet the academic standards of the organizations that accredit "regular" colleges and universities -- and that's why, in most cases, their courses/credits will not be accepted for transfer credit by "regular" colleges and universities. If you complete your basic nursing education at one of these schools and want to return to school to further your education later, as so many nurses do, you will most likely have a hard time finding a school that will accept any of your previous courses. (I notice that most of the tech/voc school websites I've looked at have the "we're accredited!" information in big letters with a flowery paragraph about how important accreditation is, in some special, highlighted section of the website -- and then there's some fine print down at the very bottom of the screen (or buried somewhere in the "FAQ" section) that says that their courses are unlikely to be accepted for credit by other schools (they're now legally required to disclose that)).
Hope that clears things up somewhat. I know it's v. hard to figure all this stuff out when it's all new to you.