fall out vs egress

fall out is not a noun while egress is a noun.

Word	Noun	Adjective	Verb	Adverb
fall out	No	No	Yes	No
egress	Yes	No	Yes	No

As verbs, egress is a hypernym of fall out; that is, egress is a word with a broader meaning than fall out:

Other hypernyms of fall out include come forth, come out, emerge, go forth, issue.

fall out (noun)	egress (noun)
	the act of coming (or going) out; becoming apparent
	(astronomy) the reappearance of a celestial body after an eclipse
	the becoming visible

fall out (verb)	egress (verb)
come to pass	come out of
have a breach in relations
drop out
leave (a barracks) in order to take a place in a military formation, or leave a military formation
come as a logical consequence; follow logically