I'm going to bold some common US psych terms in this post. This kind of confusion about what can and can't be done for/with Eugenia comes up over and over again when people are talking about her. I'm never sure how much if it might be just down to confusion about mental health treatment in the US vs in other places.
In some European countries, the government can and will commit someone with a raging eating disorder. If you've spent much time at all on the eating disordered side of Instagram, you've seen this happen countless times with the various skellies who seem to rattle in and out of treatment centers. People do get the wrong idea that this is also the state of affairs in the US, but it isn't. It's pretty impossible to commit someone in the US against their will, unless they're actively threatening to hurt themselves or someone else. And yes, I know that people will bring up "BUT THIS SPECIFIC CASE WHERE THIS HAPPENED FOR REAL"-- I'm speaking generally. If you do call a wellness check (the common US term) on someone, you'll be asked if you're having thoughts of hurting yourself or someone else. If you say no-- and someone like Eugenia (or Ash) 100% knows that the answer is "no"-- then they're really unlikely to take you in. Even if you do get taken in, you're unlikely to be held for more than 72 hours (the "72 hour hold" is the common US term for this one). Most psychiatric hospitals in the United States exist to get you stable and then to turf you right back out.
Maybe some of the confusion is the term rehab? A rehab more commonly refers to drug and/or alcohol treatment (again, in the US). Treatment for eating disorders, mainly anorexia nervosa or bulimia nervosa, generally takes place in a treatment center. Treatment centers are a big private industry in the US. Some of the big names include Eating Recovery Center, Veritas, and Renfrew, but there are a ton, and they often have multiple locations. Unlike a general psychiatric hospital, where you're treating your depressed people, schizophrenics, and so forth, treatment centers treat eating disorders and eating disorders alone.
If we're talking about rehab for drugs or alcohol, then yes, there are indeed instances where a US judge can send you to it, like it or not. That's more likely to happen if you've gotten into some sort of legal trouble involving substances, but Eugenia appears to be a law-abiding skelly. I've heard her use the word "rehab" to describe her treatment in California, but god only knows what the place was actually like.
The upshot is that Eugenia isn't doing anything actionable or anything that's likely to get her banged up against her will. I would bet many thousands of calories that her mom drilled her about what to say to first responders, after her friends succeeded in getting her put away. It's also worth noting that the friends deliberately launched that plan in California, because it's somewhat easier to get someone involuntarily committed there. Another note for the non-US readers: remember that all these laws differ, sometimes radically, from state to state in the US. It's confusing for everyone.
My own personal theory for that one has always been that they asked her some questions to assess her mental state-- like the date, president, all that stuff that paramedics are going to ask anyone-- and she wasn't able to answer those. My source for this theory is basically "I have heard this woman talk, and there's no way that she knows the date or the president."
As others said above, her mom isn't doing anything wrong unless she's actively shackling her daughter up in the attic to keep her from going to treatment, which she clearly isn't. Eugenia gets to decide whether she engages in treatment or not unless she gets a court-appointed guardian, Britney Spears-style, and I can't imagine why that would happen (unless something radically changes). We can debate all day about how competent she actually is, but US law makes it extremely difficult for a court to say she isn't competent.
My own opinion is (and has been) that she knows damn well what she's doing. Whatever pervy nonsense is going on (that leg raise she does in all her "hauls," jfc), she's aware of it. Her family does appear to be completely batshit, and her mom is absolutely an enabler, but Eugenia is living the life she wants to live.