All nursing programs are approved by their state's BON, or they wouldn't be operating. However, the proprietary (private-for-profit) tech/voc schools like ITT Tech are often not NLNAC approved (national nursing accreditation), and usually not accredited by the regional academic accreditation organizations that acredit "regular" colleges and universities (general academic accreditation, not nursing-specific). This means that the courses you take at a proprietary school usually can't be transferred into a "regular" college or university later on, if/when you want to further your education, and you basically have to start over "from scratch."
While many people start out in nursing thinking that all they'll ever want to do is basic bedside nursing and, once they finish nursing school, they'll never want to go back to school, many, many nurses find that, once they're licensed and practicing and start to see the wide range of additional career possibilities available to them with a BSN or higher degree, they start to feel differently (just look around this site at all the threads about individuals returning, or planning to return, to school). IMO, there's no justification at all, in this day and age, for paying money for academic credits that you can't transfer later to another school (esp. not for paying a lot more $$$ than necessary for credits you can't transfer!)
Also, most programs for furthering your nursing education and career (BSN completion programs and graduate programs in nursing) require that you be a graduate of an NLNAC or CCNE (the "other" national nursing accreditation organization, but they only accredit BSN and graduate programs, so that's not an issue with ADN programs) accredited program, and there is a growing trend among healthcare employers (including some of the most desirable employers in healthcare, inc. the VA, US military, and many large academic medical centers) to only employ nurses who are graduates of NLNAC or CCNE accredited programs. Choosing to attend an unaccredited nursing program now can close a lot of doors for you, permanently, for many years to come in your nursing career. IMO, it's a really bad idea, totally apart from the generally poor reputation the ITT programs have.