The easiest way to obtain a golden egg is on the Chicken Island (the middle island) in Chapter 3-1 as mentioned above.
The lower your health the more likely the game is to give you a better egg. If you have a lot of health you will most likely get white eggs over brown, and if you have very little health you will be more likely to get a gold egg over brown. It seems the game's difficulty also help to determine what eggs are dropped. So you are more likely to obtain a golden egg if you play on amateur and with as little health as possible.
Just load Chapter 3-1 on Armature difficulty with your health in the red. When you shoot a chicken it should give you a golden egg. You can also check inside the house, I have found a few of them inside.
As weird as this seems it was taken out of the Official Resident Evil Strategy Guide. It worked for me.