Does the employer have a right to deny her worker’s compensation claim for the  car accident?

Gladys an administrative assistant at a brokerage firm was involved in  a slip and fall accident at work. On her way to a doctor’s appointment  for her work-related injury, she is involved in a car accident when the  vehicle behind her failed to stop and rear ended her as she was  entering the parking lot where her doctor’s office is located. The car  accident has now resulted in significant injury to Gladys’ back. When  she arrives that day at her employer’s office, she informs them of the  accident, at this point her employer denies her of any benefits from  workers compensation.  Gladys is now required to file all medical treatment on her health  insurance and to file for short-term disability as she recuperates from  her injury and is unable to work. As Gladys attempts to recover  compensation from the at-fault driver, her health insurance and shortterm disability provider inform her that they want to be reimbursed  from anything that the at-fault driver’s insurance pays related to the  accident. Gladys is worried as she doesn’t want to lose her job, but she  cannot afford the mounting expenses.
Case Questions: 1) Does the employer have a right to deny her worker’s compensation claim for the  car accident? 2) Is it legal to submit claims under your health insurance for a worker’s  compensation injury?  3) What do you recommend Gladys do in this situation?