Getting a sick day sounds like it should be a constitutional right. Unfortunately, the federal constitution does not promise workers sick days. Additionally, there are no federal laws that require companies to give sick days (although there could be a state law). The United States is an “at will” employment country, that means the baseline rule is that anyone can quit for any reason and anyone can fire you for any reason (except for a handful of exceptions like racism and sexism).
If your employer does grant you sick days, whether you can be fired for taking one depends on the nature of your employment relationship. Most major companies will give out employee handbooks. The handbook will outline when and how you can be fired. It will also usually cover the number of sick days you can receive.
You shouldn’t, however, take a sick day whenever you want. Despite the protections offered by state law and your handbook, people still perceive how you act at work. For example, try to avoid taking a sick day before a major presentation. If you saddle a colleague or subordinate with preparing for the presentation and then swoop in to take the glory of actually delivering it, you risk alienating your team and your supervisors.
If you believe you were fired because you took a sick day, you may want to speak to a lawyer. While the baseline rule is termination at will, the reality is more complex. Furthermore, an attorney can go over your state law employment rights. You don’t want to risk surrendering your right to fair compensation for being treated wrongfully because you tried to tackle this problem on your own. A lawyer can help.