Check Overseas Births Register for army births?
Source   Gro Regimental Birth Indices (1761 To 1924)
Archive reference   REG1
Record set   British Armed Forces And Overseas Births And Baptisms
First name(s)   John
Sex   Male
Last name   Burke
Birth year   1828
Regiment   74th Foot
Type   Armed Forces
Country   Ireland
Place   Cork
Page   44
Archive   
General Register Office
Record's year range   1828
Volume   1304
Line number   102
Category   Birth, Marriage & Death (Parish Registers)
Subcategory   Civil Births
Collections from   Great Britain, UK Other
There's another one with the same details, but on page 48 instead of 44. Used findmypast for that, not sure if that's the best one to use for these records? Too bad John Burke is such a common name, it's hard to figure out who is who. This one would've been 30 when Eileen was born.
No St Aubyn Burke, Stella Burke, Eileen Gertrude Burke on any of the New South Wales BDM, newspapers, findmypast, familysearch or ancestry. Couldn't find any more information on Eileen from Paper's Past either (NZ newspapers), only two obituaries which Christine53 found.
Tried NSW BDM for just Burke and parents given names as John and Stella, no records.
Waiting on NZ death certificate. Going to continue with Australia in the hopes of finding something. Tried Victoria and QLD too. If I'm unable to find anything will post to the Australian subforum.