Nope. No rooster necessary for eggs. If you want fertile eggs and more chickens, then you need a rooster. But if you did that, about half of them would be roosters, which would be my definition of Hell. You get an all-girl egg squad by buying "sexed" chicks. Even then, you might get a rooster or two. Unfortunately, what you need to do then is to kill them. If you are trying to populate an egg ranch, you are going to have to do a lot of rooster murder.
Goats and cows are a different story: To get milk, you have to breed them. But you can use artificial insemination rather than finding a bull or billy goat. They have the calf or kid, and then you wean them when they are old enough but keep on milking mama goat or cow. They will produce milk for a few years before they dry up and you have to do it all over again.