The situation you describe is unreasonable on the part of the university, and present day politics being what they are I wonder if it's motivated by the administration's own attitudes about firearms and firearm owners in general. That said, sounds like you're asking us about what's in your contract. Shouldn't you have read it and cleared up things like cleaning equipment at home when you took the job?
If your contract specifies (and is legally valid) then obviously the answer is whatever your contract says. If you are asking whether it's okay (ethically speaking) for the contract to specify one way or the other, no it's no nice of them to ask you to do work at your own home on your own time but there's not much stopping them. Sounds like it's an unprofessional employer and maybe you could do better elsewhere.
If the contract does not specify, then it depends on the laws of wherever you are. That is also a question not for us, but for a lawyer familiar with your jurisdiction. If you want to try and sue the university to make them bend your way, go ahead and set up an appointment with a lawyer and give it a try. Beyond that, there's no useful legal advice we can give here. Except that it would probably be much easier to simply quit and go elsewhere.
If neither your contract nor the law specifies anything, generally speaking, whether you clean your firearm at home or at work makes little difference to what they have to pay for. People taking work home is common in many jobs and if they want to set boundaries on that it's basically between them and the employer. Your situation doesn't sound too different.
A very common situation is salaried employment at-will. If this is you, they can basically ask anything they want (unless illegal) and they don't have to pay for any of it. If you refuse they can fire you and they are not required to give a reason. The viable option here is to have leverage (such as being a valued employee that they don't want to let go) and persuade them to reconsider their unreasonable demands.
If you are instead hourly you probably have some sort of timesheets you fill out. You must have received instructions on how to fill those. Within those instructions you can try to simply put in the gun cleaning as additional time for which they should pay you. Maybe you can even ask for overtime pay since it must necessarily happen after work. Your employer may or may not accept this. It is up to you to negotiate or quit. That said, cleaning a gun takes a few minutes, so I'm not sure if it will amount to anything in the end.
Likewise you can also put in an expense through your HR, or whoever manages it, to compensate you for the gun cleaning supplies. Usually cleaning a gun costs pennies - a few drops from a cheap bottle of oil and a few old rags. But if you feel strongly about being reimbursed for this you can make a case to your manager. Again, they are not required to see things your way, so it's up to you to talk them into a yes, or accept a no, or quit.
More importantly is the employer now liable for any damage or injuries that may occur for a negligent discharge while at the employees' house?
IF there is no specific clause covering this, AND IF your employer recognizes this as "work" (eg. by accepting your timesheets showing "gun cleaning" at home) then they would probably carry at least some liability for accidents connected with cleaning the gun because they happened while you were fulfilling your job duties with equipment provided by them. If you are concerned you should talk to a lawyer and/or insurance carrier.
However, I find it very hard to imagine that a university would hire people to carry around guns they issue and not have stacks of liability waivers and other things that you have to sign. It's probably not a matter of general practice but a matter of what your actual contract says. Again, we cannot know that. But bear in mind also that just because they are liable doesn't mean they are at fault. With a negligent discharge, it could easily be argued that it is still your fault because you were the one being negligent. They may even sue you on top of that for damaging their reputation. It's a complicated matter best discussed with a lawyer and a deep knowledge of the applicable details.
