First, you need to determine whether your employer is subject to the Americans with Disabilities Act. The requirement is having a minimum of 15 employees and the business impacts or is involved in interstate commerce.
Assuming your employer is covered, then you have the right to ask for "reasonable accommodation." Although it is true that the term "reasonable accommodation" is subject to interpretation, I am reasonably confident that your back condition constitutes a disability and allowing you to bring your own chair (rather than insisting that the employer provide one) is all but unequivocally reasonable accommodation.
Contact the Florida Human Relations Commission, https://fchr.myflorida.com/contact/. Explain your circumstances and they can advise. If all else fails, you can file a charge with the Commission. I suspect that your employer will make the accommodation without much fuss if the employer discovers that it is obligated to do so. You could go on to the federal EEOC and even sue eventually, but you should never have to go through such hassle or expense.