I squirrel hunt all the time - with dogs and have been doing it all my life. I use a .22, but a shotgun is more efficient at actually killing them, as you will have some that will see you and run through the treetops.
A funny story... In some areas around where I live, opening day of squirrel season is a big deal... Some years ago, I was traveling with a friend and we planned to stay at his parents house on our trip - which happened to be on opening day. He told me to bring a gun, and go squirrel hunting with them, so I did - I brought a ruger 10/22 with iron sights.
When we got to his parent's place the night before opening day, about a dozen people were there spending the night to get up and hunt in the morning. They live out in the sticks, and went hunting out in some big woods behind the pasture.
I had never met these people, and when it came out that I had brought a .22, at least half of them offered to loan me a shotgun for the next morning. I told them I thought I would be OK, and they kept insisting on it, like I was some misguided and underprivileged person. I stuck to my .22 and the next morning was the only person to get my limit. They were duly impressed.
Like one of the posters above, I use a CZ452 with iron sights now, and the 10/22 is retired.