This is a good question so I am going to answer it using my experience. I believe that if you try to live without fruit and vegetables it is possible to live but you would definitely not be healthy.The reason for this is that fruits and vegetables contain more vitamins and enzymes simply because they are fresh and not processed foods.It also depends how you eat them otherwise you may not benefit from their nutrients. When eaten on an empty stomach that is when the body benefits from their nutrients most so the best time to eat them is for breakfast.
It is the same with vegetables. They also have valuable nutrients and should be eaten raw just as fruits. If they are cooked then most enzymes and vitamins will be destroyed because they are sensitive to heat. The best way to eat vegetables is to steam them. Then you will benefit from them. This way you can be sure to have a healthy lifestyle and avoid illnesses.
With regard to eggs and meat the answer is yes a person can live well without these two types of food. The reason for this is that some people have allergies to these foods. Others love eating them and do so not realising the level of cholesterol and disadvantages they have. It is also possible to live without meat as the protein we need in our bodies can be substituted by having pulses, nuts, grains, and other healthy foods available on the market now.
I have been a vegan since I was 14. This means I do not eat eggs , fish or meat and I can honestly say that my health check ups show that I am healthy every year thanks to my diet. I live on fruit and vegetables mostly and have more energy than ever before. I don't lack nutrients I need but do have one or two supplements made from organic natural ingredients which make a difference to my diet .This is a lifestyle philosophy which everyone could benefit more from. Hope this answers your question.
My own experience in life