Yes they do. Employers are responsible for what happens on their premises. If they suspect someone has drugs or is doing something illegal, even if it turns out they are wrong, they are allowed to question an employee. They are not the government so things like the right to remain silent or Miranda rights do not enter the equation. Florida is an at will state so an employee can be terminated for almost any or no reason whatsoever.
This is not to be considered legal advice nor does an attorney-client relationship exist.
Eric's answer is correct. You certainly have the right to not answer any questions and further the right to simply walk away and leave the job. I suspect that if you refuse to answer or if the employer thinks you are not being truthful, your employment will be terminated. As it relates to a claim for unemployment compensation, walking away from the job will cause you to be ineligible for benefits. If they terminate you, there should be no problem in receiving unemployment.
Yes. Your employer has that right. If you fail to comply, This is a proper reason to terminate you. Heed the advice of not quit under any circumstance to preserve you right to u/c benefits