The computer science program was strong when I was at Hampton. Good choice in schools. Because I chose hampton, it changed my life. I worked hard as a student and received a full scholarship because of connections that the department had... Got me internships, a job and I'm grateful for life. Find out which schools are accredited and which one you think you would do best in. I'm not as familiar with the business school, but I remember speaking to the department head about getting my mba there, and he discouraged me from doing it. I forgot the exact reason, but to me, it seemed he did not believe in the business school program.
The issues you mention related to your self esteem would be an issue at an HBCU or a non-hbcu if you believe those ideas in your head... I would think you'd have more of an issue being "too dark" at a non-hbcu, because at an hbcu there will be people darker than you and probably hundreds or thousands your same color. Also as for being "too fat" I would recommend 2 things. Don't worry about what other people think so much. It is important to be healthy and work on your weight to be fit and active, but if people don't want to date you for whatever reason, let that be there problem not yours. There is always someone out there for you no matter what. I'd think your odds of connecting with other folks like you would be greater at an HBCU. Lots of recruits are coming from your same background. Visit some HBCUs, email some folks from the schools you like.
Try to visit the schools you are interested in and also contact people from those schools to get the scoop. I visited FAMU and quickly figured out that it would not be a good school for me. I went to hampton and it was one of the best decisions I ever made
I went to Hampton for undergrad, computer science. When I expressed my interest in the mba program (with the business school department head) he recommend I consider another school for my MBA, mostly because I attended Hampton for undergrad... I was disappointed. I don't think I'd ever turn away a high potential student if I was a dean of a school!