The OP did have a worker's comp claim. The employer was responsible for the medical bills incurred by the accident/injury. They just did not have to pay for the lost time he/she was off work because in MA, they do not pay for missed work in Worker's Comp until 5 or more scheduled days are missed. But it was a clear cut worker's comp injury and there should of course be no disciplinary action for the absence involved.
When it comes to OSHA and required equipment for safety reasons, they do tend to get involved with correct attire, in cases of steel toed shoes and helmets and headgear, protective gloves and the like. The employer could issue standard apparel for the job if they chose to, or not.
But when it comes to say, wearing proper outdoor attire for cold weather jobs, government regulators do not tend to see this as something the employer legitimately is liable for or something they need to get involved in. I'm really rather quite pro-OSHA, because I can vaguely recall people talking about doing dangerous work before OSHA, and the spectacular lack of concern on the part of employers before they were forced into compliance by the federal government.
They're probably not going to terminate this woman for having had the worker's comp claim due to frostbite. It wouldn't be smart for them to do that. However, they are probably not going to be in any way responsible for her future resistance to cold. If she has been fully released by her doctor, and then she has another frostbite, she'll be having a whole new worker's comp claim. Reasonably, she may need to be thinking about finding some other type of job that isn't being outdoors in all weathers.