Can a company fire you if you get hurt on the job?

There is not enough information in your question for an answer. Under certain circumstances, the employer may not be able to fire you, or at least could face liability to you under one or another theory--for example, where your injury is due to an employer safety violation or clearly wrongful action. It also may not fire you simply because you applied for worker's compensation or used FMLA leave.

On the other hand, if your injury was due to your failure to obey safety rules, your carelessness, or your deliberate wrongful act, the employer most likely can terminate you, especially if your actions or carelessness threaten others.

The facts are critical to a situation like this; you should repost your question with additional detail.