Does anyone have any experience with this, or know what to do when you suspect a friend of becoming anorexic/forming eating disorders?

I'm really not sure if it's anorexia, or just the effect of a physical illness, but my friend has been throwing up a lot this week, due to the illness(flu-like?)... but what worries me most is that she seems a bit proud of losing weight from this, and will mention how she only eats one thing a day like a bagel. She doesn't binge or anything.

She's very fond of cosplay, and is always sewing costumes for anime conventions.
I remember even before she got sick, she'd often talk about wishing to be skinny like all of the professional cosplayers she idolizes. This worries me the most, because I think maybe that the image of "perfect cosplayer" and skinny anime characters is maybe getting to her head, and causing her stress, maybe she's thinking: "you can't be perfect like anime characters unless you're skinny". She is very much a perfectionist, especially when it comes to appearance, cosplay, building, and art.
She's not skinny, but she's definitely not obese, and she's very pretty. She's been healthy with self-maintenance and takes good care of her skin and such. She's popular too, with friends nationwide, so maybe that puts additional stress on wanting a better appearance...

I can understand being a perfectionist since I also am one, when it comes to art, but I also know my limits. I just can't understand the wish for a perfect body/appearance, but maybe it's because I already have a decent one, and don't deal much with social pressure/cosplay.

I am a bit afraid to ask her because it's really too early for me to be sure if she's forming an eating disorder(only been about a week) or it's just a temporary thing, and I don't want to put any ideas in her head...
For now I'm waiting for her to get better from being sick, and hoping she continues eating as usual.

My friends and I are always telling her she should eat at least vegetables and such, and suggested a Korean diet if she wanted to lose weight badly, but it doesn't seem to really get a response.

In the meantime, is there any advice, or anyone who knows a similar situation?

Well, my opinion is probably most likely irrelevant, because I have no experience with anorexics, but I can understand her wanting to have a "perfect body", I mean, who wouldn't want it?
I think the only way for you to get to her is to try to give her an a good alternative, and try to convince her that dieting in a more healthy way, will not only last longer, but will also give her better results.

Anyway, I don't know how much she weighs, but if she can lose couple of pounds while staying on the healthy BMI weight, you don't have to worry too much. I mean, I know you're worried about her because she's your friend, but I would've been angered if someone is always trying to tell me how to eat and what to eat. Especially from someone who is skinnier than me (well, it's true hehe). Perhaps it's just a phase and she'll see it's not working so well (because it won't, definitely, even if she'll lose weight, she'll gain it back straight up as long as she doesn't change her diet in general).

Anyways. Just try to give her alternatives. Also, while dieting, it is recommended to work out in order to have a leaner body and to burn "stubborn fat" that won't go away in certain places unless you burn them through high intensity workouts. But remember, 70% is the food intake, the rest is workouts. So yes, you can lose weight only through dieting, but the better and quicker results are combining it with workout.

There are many diet recipes online, she probably won't look for them by herself so just link them to her. The right amount of calories (for dieting) per day for a grown up woman is around 1200-1500 calories (depends on your height). Don't get below 1200.

Also, tell her that if she's starving her own body, her body will go into "hunger mode" and will store up fat more than usual + slow down your metabolism. Which means, if she'll continue with her hunger strike, it'll be more tougher for her to lose weight and after she finishes her diet and try to eat normally again, she'll most likely gain all her weight back.

Anyways, just try to reason with her. I'm pretty sure she knows those facts by herself...Or perhaps she doesn't as she's starving herself and eating 1 bagel a day. So remind her/enlighten her. Just telling her "you shouldn't do that" won't work.
I hope it's just a phase though, and she'll stop. However, if she doesn't, try to find a girl who was hospitalized and was an ex-ana, even online, and ask her if perhaps she can talk to her as she will understand her better than we do.

Thanks so much for the reply. I have mentioned how some starving people can get big bellies, but she most likely hasn't heard of the "hunger mode" thing, so I can try mentioning that if she brings it up again, to see if she knows or not.
So far she hasn't mentioned anything about eating, so I'm still waiting, and hoping it's just a phase.