I was not aware that it was a private university, even tho' it was probably obvious. Would it make a difference if they were at Washington State?
Indeed it would. There are many state-level laws making the carry or possession of firearms illegal or prohibited on many areas of public property, such as schools, courthouses, government buildings and the like. The same applies for many state colleges and universities. For example, there is a specific WAC for WSU spelling out firearms possession:
WAC 504-26-213
Firearms and dangerous weapons.
No student may carry, possess, or use any firearm, explosive (including fireworks), dangerous chemical, or any dangerous weapon on university property or in university-approved housing. Airsoft guns and other items that shoot projectiles are not permitted in university-approved housing. Students wishing to maintain a firearm on campus for hunting or sporting activities must store the firearm with the Washington State University department of public safety.
[Statutory Authority: RCW 28B.30.150. WSR 08-05-001, § 504-26-213, filed 2/6/08, effective 3/8/08; WSR 06-23-159, § 504-26-213, filed 11/22/06, effective 12/23/06.]