The server that the game gets its information from is http://static.monsters.tinyco.com/
Each monster has a codename that is (usually) different from its species name, and I've made a list of those in another blog post.
When retrieving an egg image, the format is 'egg_<name>monster.png' or 'egg_<name>shinymonster.png', where <name> is the codename in lowercase. Adding '@2x' before the '.png' will give you a double-size image, which is cleaner than if you just up-sampled it. For example, http://static.monsters.tinyco.com/egg_pillbugshinymonster.png
Similiarly, each monster has a thumbnail image (the normal baby form) and head images of each phase as both normal and mythic. The images of their other forms are taken from the SWF files of their animations, usually idle.
More details soon.