It is often common for the person you believe to have an eating disorder to deny they have a problem at all. This is where things get tricky. You certainly don't want to engage in a futile and often inflammatory battle of wills, but nor should you completely surrender to their denial. In learning about eating disorders, you will come to understand that this denial, and subsequent anger or refusal to seek help is a part of their illness. They will likely be ashamed of their situation and moreover, fear that by seeking help that they will lose control of their own weight.
To compromise, you might want to offer up information or point them in the direction of resources that could help them. Allow them to deal with these in their own time. If receptive, you could also offer to go to 'just one' appointment with a doctor, eating disorder specialist or nutritionist with them. This might help start to turn things around without demanding a commitment to ongoing treatment.
Where possible, focus on the person too
When someone's behaviours around food seems problematic, it can be very easy to focus only their eating habits when you speak with them. Likewise, it can be very hard to bite your tongue when someone is blatantly not looking after themselves. Understand that these problems are more about their own emotional battles and difficulties with self-worth than they are about fats and carbs. If and when you do talk about food, make sure you balance it with your concerns about them as an individual.
Be honest, be brave
There is a very fine line between being honest about your feelings relating to someone's behaviour and emotionally unloading your problems onto them. For people with eating disorders, it is good for you to model for them that it is both important and acceptable to get in touch with feelings that might be the source of shame or feel dangerous. Let them know what behaviour or action they have taken you want to discuss, how it has made you feel, and how you are going to personally deal with it. Explaining to them how you have taken the time to learn about eating disorders and just listening to them are great ways of showing them you care without making it seem like you are forcing them to change