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After a minute of research I'm inclined to believe Godzilla egg-laying only happened in Roland Emmerich's 1998 film.

Here is some contemporary reporting about it: https://www.chicagotribune.com/1998/05/19/godzilla-lays-an-egg-does-this-surprise-you/

Big, buff and bodacious, he’s so cool he can even reproduce himself–or herself. Turns out, Godzilla’s a hermaphrodite.

Consistent with the mythology, this giant lizard is a mutant by-product of nuclear radiation. As the only member of its species to have survived a bomb test in French Polynesia, Godzilla must assume male and female reproductive functions to maintain the lineage.

Why Godzilla feels compelled to travel all the way to Manhattan to lay its eggs is a mystery not clearly explained in the script, but, like any Sinatra fan, the monster probably thought, “If I can make it there, I’ll make it anywhere.” So, it was off to New York, New York, where–like the Knicks–the creature lays a lot of eggs in Madison Square Garden.

see also: https://fictionhorizon.com/how-does-godzilla-reproduce/