Should I eat eggs everyday?

Egg yolks. While the whites of the egg contains protein, the yolk contains fat so limiting the yolks at least would be helpful. If you have trouble separating the white from the yolk, shoot for no more than three whole eggs weekly. Recommendations always bounce around about this, but the fewer the better.
Not necessarily. They aren't as bad as once thought, but they're not a requirement of a good diet either. You can eat them daily. Your over-all diet is more important. Variety is the secret. Fruits, vegetables and some protein is a good start. Keep sugar and starch to a minimum. Don't leave long periods during the day without eating. That promotes fat gain. Move, walk or whatever to raise your heart rate every day.