Eggs spawn in nests around each area, next to the creature that guards them. Every area has its own pool of eight eggs, and the whole set resets every 5 minutes, so if the egg you want is not there, wait for the reset.
The reliable way to identify one is by shape, not colour. Egg colours vary between spawns and the colour does not change the income of the pet inside, so a differently coloured egg is still the same egg. Match the silhouette against the full-size image on the egg's page.
Tap any egg image on this site to open it full screen, where you can zoom in and compare the shape against what you are looking at in game. Each area page also links to the guardian you should expect to run into on the way back.