I hunt squirrels pretty much the same way I hunt grouse in the woods. Slow stalk/walk while watching for any movements or sound. I've always used a .22 rifle, never tried a shotgun. For me personally squirrels and .22 long guns just go together like apple pie and ice cream.
I've never used squirrel calls. For me squirrel hunting is all about getting out and just hunting around. A great form of just enjoying the views and sounds the forest has to offer.
Some of my squirrel guns below. Sorry for all the photos.
This is my first squirrel gun and I have owned it for over 30 years. Stevens 15a single-shot.
Mossberg 146b tube-feed. Load up 19 rounds and hunt all day.
Mossberg 152 semi-auto for those real brushy areas and quick follow up shots.
Mossberg 142 bolty for brushy areas and quick shots.
My more recent and more expensive critter getters.
Anschutz 1517HB in .17HMR for the long range shots. I don't think it is as much fun as the .22s because of the louder shots. It is deadly on rabbits though!
Weatherby xxii (Anschutz 64)... this is just pure wickedness and crazy accurate. Best trigger of all my rimfires.