@Norikosandesu,
What hasn't been mentioned so far somehow is the ironic humor associated with that sign in front of a hospital ward specializing in Obstetrics. There seems to be a deliberate intent to blur the distinction between egg as a food and the ova from a human female.
When you understand such humor subtleties, it is mildly funny. Admittedly western (American and English) humor is sort of lost where there's someone who might struggle with the language, not to mention the cultural differences in humor.
There is a certain looseness of the author of the sign where they chose to use the Egg but are using it a collective noun. That would be using poetic license. Often with humor, it can stray out of the boundary of 'proper' grammar.
Clearly they didn't mean EEG (nor was it typed as EEG), the diagnostic test electro-encephalogram, which would be EEG not EGG.