My DS (12, almost 13) stopped eating lunch this time last year, and then also breakfast. It worried me terribly. He had got the idea he was 'fat', which he wasn't and isn't.
I did talk to him about it directly; and eventually (around March, on a day when all he was proposing to eat was 2 rice cakes, an apple, and some mushroom soup made with 6 mushrooms, half an onion, stock and water :( ) I had quite a long and pushy conversation with him, and I discovered that he didn't know/understand some fundamental things about food and nutrition. He's a bit geeky :) and likes facts and information, so maybe this approach wouldn't work for every child; but perhaps sharing some of his misunderstandings will be useful for other parents:
He thought 'calories' were bad, he didn't know that they were just a unit of energy, and essential for life. I explained it in terms of him needing calories, just like rockets and cars need fuel.
He didn't have any idea how much 'fuel' he needed and how that compared to what he was eating. I got him to Google (with me) some info about recommended calorie intake for a 12 year old boy of his height. It worked out as 1500-1800 (depending on the source of the info). I then got him to work out the calorie content of what he was planning to eat, and it came to around 200 - so a shockingly long way short of what he needed. Even he could see that wasn't enough.
He didn't realise that brains need calories! He likes using his brain, so the idea of needing to 'feed his brain' (and not just his body) was attractive to him.
He didn't realise that if he cut down the amount he ate, he was also limiting his essential vitamins, minerals, etc. We talked about 'empty calories' - i.e. sugary drinks and foods that carry energy but no vitamins or minerals or protein or anything else useful - compared with 'nutritious calories'. We talked about how important these other things are, and how he needs a good mix of them to be healthy.
He didn't know about 'growth spurts'. He had a (very little) bit of 'puppy fat' in Dec/Jan which he felt bad about (I didn't know that at the time). But in February he grew 4 inches and was slim again. He connected getting slim with limiting what he ate, and didn't realise how much taller he'd grown. We talked about it (and he measured himself against me!) and we talked about how, if he kept on limiting his food, he might not get his next growth spurt (or get taller than me!)
Understanding these things helped him - he started to eat more. :)
The other thing that helped was encouraging him to cook and prepare his own meals/snacks - he particularly likes making noodle and pasta dishes, with chopped veg, cheese, etc. I am now quite happy to let him make these for breakfast if he wants - it might be unconventional, but it's healthy, nutritious and filling! :)