Urge principal to be intellectual (7)
I believe the answer is:
'be intellectual' is the definition.
(I've seen this before)
'urge principal' is the wordplay.
'urge' becomes 'egg' (as in egging someone on).
'principal' becomes 'head' (synonyms).
'to' is the link.
(Other definitions for egghead that I've seen before include "where soldiers may take a dip?" , "Intellectual (colloq.)" , "Brainiac" , "Informal intellectual of ovoid type" , "Expert" .)