There's nothing I love more than going out to bust Ca. Ground Squirrels, like twoblink has stated the things are really troublesome to land owners and they do in fact carry the plague in some areas. Please, whatever you do don't eat a Ca. Ground Squirrel, after you kill one of those things and you go up to look at it you'll see that it's just a filthy little thing with fleas jumping off of it's lifeless corpse left and right, fleas as you might know are part of the culprit behind the spreading of the plague. Minimize handling the things at all costs.
Twoblink, you didn't state if this was a grey tree squirrel or a Ca. ground squirrel. They are about equal in size but there's a difference in that the ground squirrel is a non-game animal and doesn't even have a season or a bag limit while the grey squirrel is considered a game animal and does include a season. Tree squirrels are said to be cleaner living animals and if I were to hunt tree squirrels I would be very unlikely to let them goto waste, I am very interested in finding out what squirrel tastes like. I however have no desire whatever to mess around with ground squirrels other than just rid myself of them as much as possible, they just live in their little holes and breed pestilence.
As for how tough the things are? They can certainly shake off a shot, when I'm using a 22lr. I try to limit my shots to 50 yards for real clean kills because often times if I make a gut shot the things will crawl off down into their hole and from there I have no clue what happens. You'll know if you score a head shot because they will do a "helicopter" with their tail as they go into a spasmic reflex, you always know when you score a head shot on these things no matter if it's with a 22lr. or a 308Winchester because they will roll over on their belly or roll around while "waving to say hello" with their tail. I've taken to just using a custom AR15 for blasting the things off the face of the earth when the surrounding environment permits, 223Rem with varmint bullets does the job and my gun is accurate enough for 400+ yard hits.
Anyhow, will a mouse trap work? I wouldn't push it, just go with the largest rat trap you can find. If you get a rat trap try not to handle it very much so as to not cover it with your sent, get the bait on it and set it then leave it until you see that you've nailed the wittle thing.
It sounds like there is very little food in your area and that's why it chooses your yard. In my experience these things get "edumacated" very quickly and often times learn to distrust humans very quickly if they've been pestered too much. Either that or this squirrel has been hit in the head a few too many times by your soft pellets and now it's a shade low on IQ points. Either way I suspect that nothing short of introducing this squirrel to it's maker will stop it from coming back to your yard, that or live capturing it and letting it lose in the yard of your worst enemy.
Beware of the VarmintCong.
Or else a hollowpoint in my 308 would make a nice red mist...