It looks like your friend has one option from what we've been told here. And that is to file for unemployment benefits. This will take a while, several weeks at least to begin getting benefits.
Has she been off due to illness for quite a while? Does she have some recently discovered condition that may cause them to think they need to get rid of her? They may have decided to get rid of her because she's been off, or had a health issue or just because they were in a bad mood that day, and unfortunately, it is legal for them to do so.
However, if they cannot show that they had a valid misconduct reason to terminate her, then she would very likely be able to be approved for and draw unemployment benefits while she is looking for another job.
She will file a claim for unemployment benefits, they will determine if she has enough monetary eligibility for benefits (has worked for a covered employer for long enough to have wages in the first four of the last five quarters.) If this is a yes, then they will ask exactly why she was terminated. If possible, she should print off and save the exact posting on Facebook that they claim caused the termination. If they gave her anything in writing, had her sign anything, or send her anything later, she should keep a copy of that.
They (unemployment system) will ask if she had any prior warnings or write ups for this issue. They will want to know if there were any specific company policies she was aware of that forbade employees to post comments on Facebook. She will need to say that she had no idea her job was in danger, or that what she posted on Facebook might in any way jeopardize her job.
When they have taken her claim, and she has begun to certify for weeks that pass, she will be required to register for and begin a work search. The unemployment system will contact her employer and ask them the reason why she was terminated. They will obtain both parties' version of the termination, and then will make an initial decision about whether to approve or deny unemployment benefits. Then either party can appeal the decision and if this happens, there will be a formal hearing with both parties present to present their case that the company did or did not have a valid misconduct reason to terminate her and whether she should receive benefits.
She should, UNDER NO CIRCUMSTANCES, agree to sign something saying she resigns. The employer may try to get her to do this in order to try to get out of having her receive unemployment benefits which will affect their unemployment tax rates.
In my experience, medical offices are often some of the most poorly managed places to work in the whole country, with very capricious human resources policies and practices. They may be a well run, well managed operation with clear cut policies and a well written employee handbook, or they may be a bunch of dummies shooting from the hip with this firing.
The doctor's office manager may talk to someone and be told that they have made a mistake by firing her so precipitiously, that she will probably be able to get unemployment, and they may call her back and give her the opportunity to resign "as a favor, to protect your reference" or some such baloney. She should not agree to do it. If you voluntarily resign from a job, you probably will not be eligible for unemployment benefits.
Unless they had some very clear cut policy forbidding employees to use Facebook at all, I do not see anything in what she did from what we see here that could be interpreted as deliberate misconduct. Especially since she did not mention the employer by name or give out any specific information about where she works or trash co-workers. The employer must have been looking pretty hard for a reason to terminate her. While she cannot sue them for doing this to her, she can file for unemployment benefits and use them to help support herself and family while she looks for another job.
Is your friend by any chance pregnant? Unsteady on feet and tossing cookies sort of sounds that way...