It is hard, if not possible, to help someone that doesn't want help. If anything, your friend is putting you in a rough spot because he is giving you this information and not letting you do anything with it. You are not a Professional, and even professionals can't help someone who doesn't want their help. All you can do is be a friend. Tell him that you are his friend and will help him in whatever way he asks, but that you feel he is telling you for a reason and that you feel he truly needs help (if you do in fact feel this). He may not be able to articulate why he does this. You might ask him if he does in fact want to stop. Good question, no? Then you can tell him that he needs help in doing this and again suggest talking to his doctor and asking for a referral to a therapist experienced in eating disorders. Assure him that doctors and therapists are confidential and what he talks about will remain private. Tell him they are more qualified to help him than you are. Your friend can also attend a meeting of OA, Overeater's Anonymous which is free and for people with all kinds of eating disorders; there are meetings in all large towns. He can just sit and listen and not talk if he doesn't want to. You can offer to go with him. Good luck.
Wow it seems like your friend is struggling a lot, and now I get the feeling as though you are carrying this burden too.
I think it is important to recognise that your friend has already come a long way by just disclosing this information to you, but in doing this he has also signed himself up for a concerned person to be on their back!
In around a month, your friend has gone from throwing up everyday to a couple times a week - and while this may seem extremely disordered from your perspective, it is an improvement from where he was, so maybe allow him to see this as a small achievment? The only thing that sounds concerning to me is that he has now began to skip break times, which is what happened to me. I personally have gone from bulimia to anorexia (which is common) so maybe keep an eye on this as best you can.
I feel that it is important for you to realise, that while your friend has choosen to confide in you about certain things, you should not feel responsible for them, or burdened by their problems. You seem to be there for a support when he needs it (which is what most people would LOVE!) the key thing is just to be there, encouraging him - not forcing him - to practice more ordered behaviours, and possible speaking to family or doctors about it. Unless it becomes extremely life threatening(which is dependant on how long this has been going on)
For the next little while I would say just keep a watchful eye on your friend, but do not feel guilty or annoyed if he does not follow your advice. An eating disordered mind is not one that likes to listen to others advice or opinions! What they say, goes...but just keep trying and ensure that he knows you are there when necessary. Maybe offer to sit with him if/when he tells family or mention that you would accompany him to the doctor if need be.
I think that you must be very special to him for him to confide in you, but remember for him even that took time, as will this.
I hope this has helped!