I don't know anything about food phobia so I'm not entirely sure, but it certainly looked like a reaction to the mother.
It was the usual Supernanny stuff: calming the whole family situation down with naughty step and ground rules incl no shouting. Then re the food, she made them all change to eating healthier food, instituted reward chart place mats so the kids got a sticker for eating each part of their meal(she explained the different food groups to the kids - starch, protein, veg etc) and got the parents to not pressurise the boy to eat - it was entirely down to him whether he ate and got the sticker or not, and they had to lighten up the tone of mealtimes by not talking about it. She also made them offer a choice of 2 healthy menus for the kids to choose from (giving them back a sense of control), got the boys cooking with the dad and gave the older one a tomato plant to look after.
Honestly it was incredible how quickly the food problem got sorted out once the mother stopped yelling, which is why I suspect it wasn't true food phobia.