P.S. "How would you like your eggs to be cooked?" might mean they want you to say how soft or how hard you want them to be. If you order scrambled eggs, there is no problem--you don't have to say. For other kinds, you can say "fully cooked" or "runny." A more complete phrase would be "I want the yolk fully cooked, all the way through" or "I like the yolk to be runny." But "fully cooked" and "runny" will be understood. "Runny" means that when you stick a knife or fork into the yolk, and break the membrane, liquid yellow yolk runs out.

The most common choices are 1) scrambled, 2) hard-boiled, 3) soft-boiled, 4) fried, sunny-side up, 4) fried, over easy. Joe's picture is great. To make this a language exercise, I will describe them in words. In a restaurant, if you are not sure, order scrambled eggs. It is a safe choice. They are easy to fix. Any restaurant will do a good job. Scrambled eggs have the yolk and white mixed up before they are cooked. They look like lumpy yellow oatmeal. They can't be cooked wrong. Everybody likes them. Nobody hates them. Hard-boiled eggs are served cold. They are whole eggs. You crack the shell. You peel off the shell. You have a smooth, white, rubbery egg. When you bite into it, you see the yellow yolk inside the white. They are great for picnics. In the USA mothers put them into lunches for children to take to school. They keep for hours without refrigeration. You can pick them up in your hand and eat them. Soft-boiled eggs are served hot. They are served in the shell. You crack the shell carefully. Inside, the egg is cooked, but it is soft and almost liquid. You eat them with a spoon. Restaurants have trouble cooking them. They have to be cooked for exactly the right amount of time. Everyone likes theirs differently. Some people like "three-minute eggs," boiled for three minutes and very soft. Some like "five-minute eggs," which are almost hard. A fried egg is cooked flat in a frying pan. It comes out flat like a pancake. It is almost a disk, perhaps 10 cm. across. The yellow yolk is in the center, with white all around it. "Sunny-side up" means the egg is not turned. Usually, the yellow yolk is soft and liquid. People who want their egg yolk fully cooked ask for the egg to be fried "over easy." The cook uses a spatula. The cook flips the egg over once. Both sides of the fried egg get cooked.

Here is a nice link with pictures, explaining all of the options: https://www.thespruceeats.com/how-to-order-eggs-at-brunch-4176585