Bird wore no eggs out (4)
I believe the answer is:
wren
'bird' is the definition.
(wren is a kind of bird)
'wore no eggs out' is the wordplay.
'eggs' becomes 'oo' (visually resembles two eggs).
'out' is a deletion indicator.
'wore'+'no'='woreno'
'woreno' with 'oo' taken away is 'WREN'.
(Other definitions for wren that I've seen before include "A small garden bird" , "Former RN woman" , "Small British bird" , "Small bird (in the Navy?)" , "Small singer" .)