Nothing in the law forbids that. So… you can, but the question is whether you should.
First, when people eat out, they rarely complain about leaving a 15% tip, if the waiter has done an adequate job taking the order, bringing the food, and re-filling tea glasses. For that same 15%, you can hire a lawyer to render advice on the fairness of your proposed settlement.
Second, the other side is hiring someone who has graduated from high school, college, and law school, and who has passed the Bar Exam, and who has achieved at least 12 hours annually of continuing legal education; the other side will pay that person around $175 per hour to protect it. And the other side is a corporation which has a legal duty to its shareholders to maximize its profits by keeping money coming in much higher than that which is going out to you and others. Is that a level playing field?
Finally, consider this: that high-priced suit hired by the other side spends a lot of time drafting paperwork – which you will be required to sign – that protects his client. Are there provisions in there that will give up your rights to medical treatment? Are there provisions that will forfeit your rights to any other claims you have – whether you know about them or not – relating to a variety of rights such as fair pay, proper overtime pay, and the like? Is there language that jeopardizes your job security, or which gives away your rights to further compensation if your injury worsens and you are unable to work further?
So, yes, you can legally settle your case without a lawyer. But should you?