No. You should not share information about your accident on social media. While it may serve as a way to update multiple friends and family at once, it could come back to bite you. Unfortunately, a defendant or their insurer might try to twist anything you post publicly against you.
Photographs of the accident scene, information about your injuries, and other shared details may land in the wrong hands. An insurance company may use your post or anything related to it against you to blame you for the accident or downplay your injuries.
Limiting yourself to telling about the accident verbally to your closest family is generally a good idea to start, and one that can protect your case.