Your friend said that if you eat too many eggs too frequently, you can gain an allergy to them. Myth. This doesn't hold the slightest truth to it in any way what so ever. There are, however, bad things about eggs.
Many people are under the impression that you can live off eggs and that they're the most healthy food on the planet. To some extent, they're right. Eggs are an extremely good food for you. There are negatives of eggs too, however.
Eggs contain large amounts of cholesterol. In every egg there is 212 milligrams of cholesterol, which is quite an alarming amount. This is rivaled by only a single server of liver, shrimp, and duck meat. For you, as a bodybuilder, having slightly higher cholesterol levels than everyone else isn't so bad, as your body requires higher levels of everything just to prevent it from shutting down.
You also said you were bored of eggs. Try cooking them differently! There is a huge range of different recipes with eggs as the main ingredient. Changing the way you cook your eggs can be one of the best ways to overcome this boredom. Have a look at some different recipes rather than just eating them by themselves. Perhaps if you incorporate other necessary foods as the primary ingredient and the egg as the secondary, it'll change the way you feel about eggs.
If you don't want to try that, there are alternatives. The alternatives are never going to be as energy packed as eggs are, but they should still offer similar amounts of protein. You can try red meat. A steak sandwich for breakfast is a great way to load up with iodine and it contains a lot of protein. Perhaps you'd like to consider a complete meal replacement or two? High protein meal replacement shakes are a good way to go, however I think you'll get bored of these quicker than you did the eggs. If you choose this option, you're going to want to look at the shakes/supplements which have higher levels of carbohydrates and micronutrients. Breakfast is one of the most important meals and if you have something like a Whey Pure Isolate (WPI), than you're not going to get the other essential nutrients your body requires for the day. Sure, Whey Pure Isolates contain loads of protein, but you're after a balance for breakfast. Look primarily for high carbohydrates, then your protein, and then fats and other micronutrients when purchaseing your meal replacement.
Hopefully, I've shed some light on your predicament and if there is anything you'd like me to go into more detail about, don't hesitate to send me a Private Message. I'll be able to go into more detail about anything you'd like to hear more about.