Quand j'étais né is impossible in French.
The root cause of the confusion many English speakers have with être né is the fact that there is no verb for naître in English.
"Born" doesn't translate with né because "born" (from "to bear") really means enfanté/accouché. "I was born" actually means j'ai été enfanté.
Être né is an event providing a status you gain at birth. Either that event never took place:
Il n'est pas né. (Il n'est pas encore né.)
or did happen (passé composé):
Il est né. (Il est né hier. Il est né le 15 août 1769.)
and when someone was born, it can't be undone.
The passé composé has superseded the passé simple in most current French usages, outside a literary context. Historically, passé simple would have been used when the birth didn't happen in the recent past:
Il naquît le 23 août 1754.
Using the plus-que-parfait like in quand j'étais né would mean you used to be born in the past but you are not born anymore which definitely makes no sense (unless maybe if this happens in a sci-fi novel about time travel and/or parallel worlds).
French uses the same form with verbs referring to the birth and death of people:
Il est né un lundi - Il est mort un mardi.
Il naquît en France - Il mourût à l'étranger.
while English is not symmetrical, lacking a verb for naître:
He was born on a Monday - He
was dead on a Tuesday
bore? on a Monday - He died on a Tuesday
PS: a newborn doesn't eat vegetables yet.