Upon further research, I've found that USU does not have the authority to limit in any way the possession of firearms at USU. The following code explains it. (critical parts are red)
Utah Code 53-5a-102. Uniform firearm laws.
(5) Unless specifically authorized by the Legislature by statute, a local authority or state entity may not enact, establish, or enforce any ordinance, regulation, rule, or policy pertaining to firearms that in any way inhibits or restricts the possession or use of firearms on either public or private property.
It goes on to further define the following
(6)(b) "local authority or state entity" includes public school districts, public schools, and state institutions of higher education.
Utah Code 53B-3-103. Power of board to adopt rules and enact regulations.
1) The board may enact regulations governing the conduct of university and college students, faculty, and employees.
(2) (a) The board may:
(i) enact and authorize higher education institutions to enact traffic, parking, and related regulations governing all individuals on campuses and other facilities owned or controlled by the institutions or the board; and
(ii) acknowledging that the Legislature has the authority to regulate, by law, firearms at higher education institutions:
(A) authorize higher education institutions to establish no more than one secure area at each institution as a hearing room as prescribed in Section 76-8-311.1, but not otherwise restrict the lawful possession or carrying of firearms; and
(B) authorize a higher education institution to make a rule that allows a resident of a dormitory located at the institution to request only roommates who are not licensed to carry a concealed firearm under Section 53-5-704 or 53-5-705.
Some may then say that because of the landlord/tenant relationship residents have with USU they can restrict it, but this code states that USU can not restrict firearms.
76-10-530. Trespass with a firearm in a house of worship or private residence -- Notice -- Penalty.
(1) A person, including a person licensed to carry a concealed firearm pursuant to Title 53, Chapter 5, Part 7, Concealed Weapon Act, after notice has been given as provided in Subsection (2) that firearms are prohibited, may not knowingly and intentionally:
(a) transport a firearm into:
(i) a house of worship; or
(ii) a private residence; or
(b) while in possession of a firearm, enter or remain in:
(i) a house of worship; or
(ii) a private residence.
(2) Notice that firearms are prohibited may be given by:
(a) personal communication to the actor by:
(i) the church or organization operating the house of worship;
(ii) the owner, lessee, or person with lawful right of possession of the private residence; or
(iii) a person with authority to act for the person or entity in Subsections (2)(a)(i) and (ii);
(b) posting of signs reasonably likely to come to the attention of persons entering the house of worship or private residence;
(c) announcement, by a person with authority to act for the church or organization operating the house of worship, in a regular congregational meeting in the house of worship;
(d) publication in a bulletin, newsletter, worship program, or similar document generally circulated or available to the members of the congregation regularly meeting in the house of worship; or
(i) in a newspaper of general circulation in the county in which the house of worship is located or the church or organization operating the house of worship has its principal office in this state; and
(ii) as required in Section 45-1-101.
(3) A church or organization operating a house of worship and giving notice that firearms are prohibited may:
(a) revoke the notice, with or without supersedure, by giving further notice in any manner provided in Subsection (2); and
(b) provide or allow exceptions to the prohibition as the church or organization considers advisable.
(4) (a) (i) Within 30 days of giving or revoking any notice pursuant to Subsection (2)(c), (d), or (e), a church or organization operating a house of worship shall notify the division on a form and in a manner as the division shall prescribe.
(ii) The division shall post on its website a list of the churches and organizations operating houses of worship who have given notice under Subsection (4)(a)(i).
(b) Any notice given pursuant to Subsection (2)(c), (d), or (e) shall remain in effect until revoked or for a period of one year from the date the notice was originally given, whichever occurs first.
(5) Nothing in this section permits an owner who has granted the lawful right of possession to a renter or lessee to restrict the renter or lessee from lawfully possessing a firearm in the residence.
(6) A violation of this section is an infraction.
76-10-500. Uniform law.
(1) The individual right to keep and bear arms being a constitutionally protected right, the Legislature finds the need to provide uniform laws throughout the state. Except as specifically provided by state law, a citizen of the United States or a lawfully admitted alien shall not be:
(a) prohibited from owning, possessing, purchasing, selling, transferring, transporting, or keeping any firearm at his place of residence, property, business, or in any vehicle lawfully in his possession or lawfully under his control; or
(b) required to have a permit or license to purchase, own, possess, transport, or keep a firearm.
(2) This part is uniformly applicable throughout this state and in all its political subdivisions and municipalities. All authority to regulate firearms shall be reserved to the state except where the Legislature specifically delegates responsibility to local authorities or state entities. Unless specifically authorized by the Legislature by statute, a local authority or state entity may not enact or enforce any ordinance, regulation, or rule pertaining to firearms.
I've highlighted a bit to point out the critical parts of the text. I think that it would be appropriate to change the text in the student housing handbook, as it is suggests that students are not allowed to possess firearms on campus.