I would encourage him in his vegetarianism, personally. If he really doesn't like meat, which was the case for me as a child, then his options are to eat vegetables, beans, legumes, breads, rice, potatoes, and dairy if he likes milk/yogurt/cheese. You mentioned he doesn't like eggs, and that's okay. If he likes salad, then he should be offered a variety of dark leafy greens, like spinach, and not just iceberg lettuce. Shredded carrot, shredded cabbage, shredded beets (if he likes sweets, then the sweet starchy vegetables like beets, sweet potatoes, peas, corn, carrots, etc. may appeal to him). Add cheese or avocado to his salad, chick peas, beans, cottage cheese, sweet peas. Serve with a side of bread or a baked potato, you can alternate between a regular russet potato and sweet potato. You could mash it with a bit of milk.
Roasted vegetables make good fillers for wraps and sandwiches. I used to eat Cream cheese sandwiches with minced apples and walnuts on a multi-grain bread. If he likes roasted vegetables, you could saute some green/red peppers, onions, and mushroom, add some mozzarella cheese, and put it into a wrap. Have you tried veggie burgers? Morningstar farms makes tasty ones and they offer those at Burger King if you're looking for a fast food option when you're on the go. You can make sweet potato or regular french fries in your oven and serve with a veggie burger. Pasta with vegetables and tomato sauce, or pureed vegetable sauce in a tomato base (whatever vege you like, maybe zucchini). You could try and offer a meatless meatball.
the only thing that will get him to like different vegetables, is exposure. Keep offering them to him, offer them to him in different ways--raw, cooked, in a sauce, with a dipping sauce, pureed, in a hummus or bean dip, in a guacamole, etc. Pair them with foods you know he likes, like fruit with cottage cheese, etc. Give him a choice--you can have a salad with x, y, z or you can have a pasta with a vegetable and tomato sauce but you cannot have plain pasta with garlic salt. You can have another piece of bread if you eat three bites of your vegetable, that sort of thing. Be enthusiastic about the foods--if you eat them, he will be more inclined to at least try them. But DON'T push. If he is distressed by an item, like turkey, egg, or broccoli, don't force it on him. Acknowledge that he just doesn't like it and move on to something he does. Offer two new things at dinner and one thing you know he likes. He'll work out what he likes and find balance. In the meantime, give him a multivitamin.
Good luck!