I answered your main question here:
Assuming that you were applying for a job with LexisNexis, you would be filing a civil lawsuit against the company. However, it is not clear what your grounds would be. As a private employer, LexisNexis is under no obligation to accept the information you provided and is free to make whatever determination it wants. More generally, LexisNexis was under no obligation to hire you.
It would be a different matter if you could show that your condition constituted a disability and that LexisNexis was discriminating against you on the basis of your disability. However, from the facts that you have stated, this does not appear to be the case. Consequently, you have only the limited rights of an at-will employee, which are discussed here: