虫 at the bottom marks 蛋 as biology: the package every small creature hatches from. The character is young by Chinese standards — a latecomer that displaced the older 卵 in everyday speech because eggs, like the word, are daily goods.
茶叶蛋 tea eggs perfume every convenience store; 番茄炒蛋 tomato-scrambled-eggs is the first dish every Chinese student abroad learns to cook — the taste of home in eight minutes. 蛋 also does insult duty: 笨蛋 'stupid egg', 坏蛋 'bad egg', 完蛋 'finished egg' (doomed).
The round bundle 疋 every creature 虫 starts inside — an EGG.
脸红得像鸡蛋 isn't a thing, but 滚蛋 'roll, egg!' is — the rudest way to say 'get lost' involves telling someone to roll away like an egg.