So ignore the idiots that suggest adding meat like it's gods gift to nutrition (honestly, how dumb can you be). If you fed him meat then you could make him seriously ill because the body doesn't always retain the ability to digest it. Regardless of the ethical question it raises. Pickiness can be all sorts of things as a previous poster has suggested and sometimes they can be due to deficiencies, put that part on the parents to investigate.
The basics of a vegetarian diet are protein comes from sources like tofu/soy, eggs, seitan, beans, legumes, cheese, nuts, seeds and vegetables. These also cover fibre, what you have to be more aware of is the grams of carbs for every gram of protein ratio, as it can be (but isn't necessarily) higher than directly swapping meat.
What you need to do is determine what is driving the fussiness. Is it flavour or texture?
Are there any swaps you can do - like increase protein by changing the yoghurt to greek yoghurt (like Fage) or even soy yoghurt?
Can you make yoghurt based dips that he'll eat (Mediterranean cuisine is great for recipes) these may go better than hummus which can be grainy and serve that with grapes, toast sticks etc?
Egg meals are great, doesn't matter if they're his daily lunch. Can you add in things like mash potato into the omlette?
Then there's experimenting with the sauces, see what he likes as a flavour and then what you can add to it to help improve nutrition. Blended lentils or beans make great thick sauces, if he'll eat mac and cheese, can you blend some butter beans into the cheese sauce? Or half the pasta and add butter beans?
Then it's the usual trick of hiding veggies in.
If he's gone off lasagne, deconstruct it to see what flavours or textures he's gone off. If you're using quorn instead of mince, try a different protein source.
He doesn't know meat textures and flavours so you don't need to use substitutes or flavour things that way.
Good luck and just keep experimenting.