If a company provides health care coverage, and one of the requirements they have for employees is to either be covered by that benefit or show proof of other covereage, and you refused to do so, then yes, they can fire you for failing to comply with a company requirement. It is usually a condition of employment that you follow the rules set forth by the company, like being on time, following the dress code, following safety procedures, etc. I don't see the ACLU as wanting to look into this, as there is no violation of free speech, or any other constitutional issue.
It seems to me that you were not fired for refusing to provide the insurance company name, you were fired for refusing to follow company policy.