I'm also having to address increased rock squirrel activity and population around my home.
#1: Killing one with something like your Sheridan is doable. That is what I used to kill one.
However, since running off other rock squirrels, they have become more wary than that first one, which sat eating on the ground, totally unconcerned that there I was aiming a big brown stick at it from no more than 25 ft away.
The big problem is not which gun, but how to get kill a squirrel that is not high up in a tree (rock squirrels both burrow in the ground AND climb trees), or running.
#2: Therefore, I think setting a baited Havahart trap is probably easiest, and that is what I just bought. It cost about $41 at the hardware store. Another $60+ for a multipump gun will allow part 2 of the plan: killing the trapped squirrel. This way, power for killing from more than very short distance is not a concern. All your friend needs to learn is how to set the trap and operate the gun in the most basic aspects.
#3: The suggestion to trap it and drown is an option if she doesn't want to use a gun, period.
The need for an aired-up gun ready to use on short notice becomes irrelevant once you use a trap. The total expenditure could be as low as $41 plus a few bucks for a large, deep bucket. Unless she WANTS to learn to shoot, there is no reason to make this more complicated than that.