This term meaning to encourage or incite has nothing to do with throwing eggs. The verb egg dates to 1200 and is from the Old Norse eggja. From the Trinity College Homilies, c.1200:
Alse the deuel him to eggede.
(And the devil egged him.)
The phrase to egg on is 16th century. From Thomas Drant’s 1567 Horace His Arte of Poetrie, Pistles and Satyrs Englished:
Db, Ile egge them on to speake some thyng, whiche spoken may repent them.
(Source: Oxford English Dictionary, 2nd Edition)
Copyright 1997-2017, by David Wilton