Efw: For the last 59 years or so I have been shooting Squirrels (mostly Ground Squirrels but some Tree Squirrels as well) and I recommend you reconsider your aversion to "bull barrels and laminated stocks"!
Both of those type accessories to your Rifle will enhance and improve your Squirreling.
Most all factory 10/22's do indeed need trigger improvements and I heartily recommend that.
Best of luck in your build.
Hold into the wind
VarmintGuy
P.S.: If you do reconsider the "bull barrel" I highly recommend the Dan Lilja custom heavy stainless in 20" or 21" configuration.
Mine, on my Deluxe 10/22, has served me EXTREMELY well for more than 25 years now and far in excess of 100,000 rounds down the bore.
