Synonyms for Egg on:

n.

inflame, goad, foment, cause, galvanize, inspire, pique, provoke, impel, stimulate, touch off, excite, spur, trigger, work up. egg on (noun)
prod, incite.
encourage (noun)
urge, goad, drive.

v.

contact (verb)
prod, incite.
push to do something (verb)
incite, urge, excite, pique, spur, prod, stimulate, drive, goad.

Other synonyms:

trigger, stimulate, foment. galvanize, impel, excite, pique. inflame. provoke, inspire. motivate
touch off.
Other relevant words:
inflame, work up, spur, trigger, stimulate, inspire, pique, touch off, goad, drive, foment, excite, galvanize, urge, impel, provoke.