Its an oval or round object laid by a female bird, reptile, fish, or invertebrate, usually containing a developing embryo. The eggs of birds are enclosed in a chalky shell, while those of reptiles are in a leathery membrane.
https://youtu.be/PL9iMPx9CpQ?t=25
See above. If you're looking for the Tumblr version, which I assume is what this is referencing, an "Egg" is their cutesy way of referring to someone who's in the closet about being Trans or something to that affect.