The "more meat = greater height" works only if they were getting insufficient nutrients prior to "more meat". What also happened after WWII in Japan (after reconstruction) was an increase in the average wealth of the people. And, on average, most people in industrial nations have been getting a little taller every generation as health standards improved, anyway -- childhood illnesses can stunt growth as well. It's hard to pinpoint one cause, especially in a correlation study (which only hints at a possible cause, requiring further investigation). The thing is, it's easy to get your minimum protein and some other nutrients if you have some meat (or fish) in your diet. If you remove that, you have to be a little more conscious of what you're eating in the course of the day. If you're going vegetarian or vegan, it's not just about what you subtract -- to be healthy, it's very important what you add.
But keep in mind, people who eat a lazy diet with meat are also often deficient in other nutrients which are easily found in lots of things vegetarians and vegans already add into their diets. Just for fun one day, you should write down everything you eat, and then the next day, look up the nutrient levels of everything and try to figure out if you got enough vitamins and minerals. Then, next time you stop at 7-11 for a soda, think about getting a V8 strawberry-banana, or some other veggie/fruit juice, instead. Or if you want a sandwich from a deli, see if they can put mesclun greens and/or spinach on it instead of iceberg lettuce, and see the nutrient difference with those choices. Meat-eating people questioning vegetarians or vegans are so quick to point to the "protein problem" while unaware that they, themselves, are likely low in vitamin A, E, etc.