As soon as she is released by her doctor to return to work, she needs to file for unemployment insurance. (Or if that is going to be quite a while, she needs to go on and file immediately. She won't be able to begin drawing yet, if she's not released by the doctor, but she needs to get the claim set up before the quarter changes again in October.
She will be filing a claim based on the wages she has made during the last 18 months to two years, regardless of what employer(s) she worked for or what the circumstances were under which she left them.
They'll determine, when she files, whether she has enough covered wages in the last 2 years to set up a claim monetarily.
If she hadn't worked anywhere for a long time before she took this job, then she'll not have a claim to set up. But if she has been working, and had say, left another job to accept this last job, she very well may have enough money to set up a claim.
Then they'll look at whether she is out of work through no fault of her own. Since she had to miss work due to having to have emergency surgery, they will determine that she was terminated for a reason beyond her control. If she's monetarily eligible, with the appropriate medical statements completed by her physician, when she is ready to look for other jobs, she'll be able to start receiving unemployment benefits until she finds another job.
Yes, an employer can fire you if you're not covered by FMLA. But they cannot keep you from receiving unemployment benefits if you are otherwise eligible, because the system says that you could not control being sick, therefore you are out of work through no fault of your own.