If you like eggs than feel free to eat them. People do argue the finer points about how healthy eggs are but unless you eat a lot of them you probably don't have to worry.
There is no good objective scientific evidence that high protein diets help any type of athletic conditioning. Repairing and building muscle does require protein so athletes do require more protein than sedentary people but since nearly everyone consumes several times as much protein as we need it isn't a problem.
Our mainstream protein recommendations come from early flawed studies and this thinking has never been revised because of financial interests of the chicken, beef, pork, dairy, and nutritional supplement industries. Consider that there is no time in our lives when we grow and build muscle faster than we are babies. Human breast milk only gets about 10% of its calories from protein. Scientists testing the nitrogen balance of human diets found that adults do not loose protein from energy sufficient diets until they get less than 5% of their calories from protein. For a safety margin this should probably be doubled to 10%. Most people easily get several times that percentage of protein from their normal diets without trying and unless you eat a pathological diet of only fruit it would be impossible to consume less than 10% of your calories from protein. Considering that athletes eat more food than sedentary people there is no need for most athletes to adjust their percentage of protein upward.
Some athletes have always gone against the grain. Multiple Olympic and world champion Track & Field star who has held the indoor long jump record since 1984 Carl Lewis was most competitive when he was a vegan. There are lots of other top athletes including champion power lifters, gymnasts, and bodybuilders who never go near egg whites, whey power, or chicken breast.
Yes eggs have a lot of good protein that will help your muscles recover fast. But you can get that from other food sources too, you can cook up a boneless skinless chicken breast it will have twice the amount of protein that two eggs have and it has almost no fat, it just takes longer, that is why eggs are so good for working out, there fast and easy, you can just boil a bunch for the week toss them in the fridge and eat two after each workout
Adding to all that you have said. You don't need to get protein in through meat, it is actually not good for you at all. You can rather get it in through certain vegetables, such as asparagus, beetroot, broccolli, rice (I know not a vegetable) etc. If you do eat meat, then eggs are the source with the most protein in.
legumes or pulses are the richest plant source of protein.
beans, peas, and lentils are all in this family.
Legumes can actually rival meat in protein density. Since many kinds of meat are fatty you can often get the same amount of protein from legumes as you would from the same number of calories of meat.
No necessary but eggs are a good source of protein and other nutrients, helps your body recover fast, but as long as you're getting these nutrients from other foods you don't NEED to eat eggs everyday, but they taste great and there easy to make which makes it ideal for an after work out snack
Before and after a workout I like to use a whey protein shake, I also eat lots of bananas for the potassium, this will help keep your muscles from cramping up after a hard workout.
The instructor in Gym always says that those who work out heavy will have to eat 2 eggs daily to gain strength as egg has got enough protein to help out for work outs