It sounds to me as though these are not the kind of people you would want a career with anyway. Just because they "could" fire your daughter, does not mean you should consider that a bad thing or that you should be afraid of them. If you have the option financially speaking, your daughter should write a detailed resignation letter explaining what she was told, why she feels it's wrong and unprofessional, and personally hand it to the owner of the company, not her boss. The reason being, I doubt the owner of the company is aware of what her boss is telling her. Just because her boss is in charge of the employee's, does not mean her boss is acting in the owners best interests or that the owner approves of her bosses actions.
Your daughter should let the owner know that she's leaving because she was specifically told there would be consequences for not taking her resume off career websites, and that she had no intention of leaving the company until being treated so unprofessionally. Explain that whether consequences meant getting fired, or consequences meant not being able to advance in the company, that all possibilities indicated that leaving the company would be in her best interests. She should explain that she wants to work for a company where she can advance, and does not have to worry about getting fired because something rubs her boss the wrong way. I wouldn't expect the owner to try to make it right in any way, but her boss will likely hear from the owner about it asking why he/she did it.
The owner likely wants to keep good hardworking employee's on staff because it helps his/her bottom line. The owner also does not want employee's (her boss included) who will open him/her up to liability. If she is a hard worker and this whole thing is out of spite from her boss, she may even be invited back with an apology from her boss, who knows. But I'd tell them to pound sand and find a company who wants to take care of her needs, not treat her like a sheep.
The truth is, the boss could of, and should have asked "Are you happy here? We noticed you are putting your resume out there, and we want you to know that we value you as an employee."
FYI: I once submitted a similar resignation letter, and I got a call from the human resources manager offering me more money to stay on. I explained to her that just like they had had plenty of time to see my worth, I had had more than enough time to see the company's worth. I said I was looking for a company to devote 30 years of my life to, and that I didn't want to devote that kind of time to a company where I would be unhappy with my co-workers while scratching and scraping my way to a meager salary. I also said that it took me quitting to finally get your attention, and that they had lost my loyalty. I fired them. My point is, make sure she knows her worth, and don't settle for any less than she deserves. In the longrun she wants to be happy, and she'll be glad she stood up for herself.