Ikuhara had said it would be better if we didn't know, but I want to know. What is the meaning behind Nanami's egg? Especially when she wakes up from an apparent dream, and finds a smashed egg in her bedsheets.
That episode was . . . weird, as weird as when Nanami became a cow.
Anyway, my interpretation of it was based on the fact that the episode "Nanami's Egg" falls into the Akio Ohtori Saga. This is roughly two thirds of the Apocalypse Saga. The content is more sexually suggestive in this saga: scenes involving Akio's car seemed to always be sexually suggestive, with his, Touga's and Kyouichi's shirts opening partway through the ride or when posing for photos, and Kozue's first sexual experience was in the back of that car with Ruka Tsuchiya.
Because of this I got the impression that the episode represented Nanami entering puberty as we know that women to begin produce eggs during puberty. She begins to fret over the discovery of an egg in her bed probably because she hasn't been taught about puberty. When she accepts it and raises the egg, I took this as the transition of Nanami growing up from a girl to a woman/mother.
Touga says to Nanami, towards the end of the episode:
Do you know why we've been able to live together so happily? It's because you aren't the type of girl who lays eggs. I pity the family of a girl who would betray them like that.
So it can't be that the egg is simply an indication of the beginning of Nanami's menstrual cycles: there's nothing socially unacceptable per se about girls having periods. Rather, Touga's comments suggest "deviant" behaviour. Presumably this is something sexual: Nanami produces an egg, and Anthy "wonders who the father is", after she says that Nanami might have laid an egg.
Thus, it's possible that the egg takes on the meaning of an undesirable pregnancy: Nanami feels forced to abandon it, and the egg is found shattered one morning.
However, the concerns Nanami picks up before she talks to Touga can evoke anxiety over puberty.
She wakes up in bed one day and notices an egg---similar to how a girl might notice her menstrual blood out of the blue. (This could perhaps be similar to the discovery of a pregnancy, but it wasn't an immediate connection for me.)
Nanami also doesn't know what is going on and is as a result scared. I'm not sure if ignorance about menstruation was common in the 1990's though.
Nanami becomes convinced that it's normal for girls to produce eggs. She then worries that she'll be mocked for not laying an egg earlier. When Nanami mishears Juri's words and believes Juri has produced eggs since the age of seven, she thinks Juri is very "mature". This parallels the linkage of puberty to a desirable sort of maturity and anxieties about the normalcy of one's body.
However, this could just as easily be about sex (rather than puberty). (Having sex can be associated with maturity; one might feel abnormal for not having had any.)
I would avoid trying to impart too much meaning to the egg (since there's nothing that entirely fits), but we can thus at least safely say that it is linked to sex (and being initiated into it) and possibly puberty. (Note that sex and puberty can be linked. For instance, one is "mature" not just for having a developed body, but also for having sex.)