hire vs sign up

hire is a noun but sign up is not a noun.

Word	Noun	Adjective	Verb	Adverb
hire	Yes	No	Yes	No
sign up	No	No	Yes	No

As verbs, sign up is a hyponym of hire; that is, sign up is a word with a more specific, narrower meaning than hire:

Other hyponyms of hire include featherbed, fill, engage, ship, contract, sign, sign on, rat, farm out, job, subcontract.

hire (noun)	sign up (noun)
the act of hiring something or someone
a newly hired employee

hire (verb)	sign up (verb)
engage for service under a term of contract	join a club, an activity, etc. with the intention to join or participate
engage or hire for work	engage by written agreement
hold under a lease or rental agreement; of goods and services