I've got a Winchester Ranger Compact 20gauge that I adore. Works well on squirrel out to 75 yards using Remington Game Loads, 7/8oz #7 1/2 or #6 shot.
I've also got 4 NEF single shots (actually 6), 28" full choke 12ga, 28" modified choke 12ga, 26" full choke 20ga, and a 26" modified choke 20ga. The 12ga mod and 20ga full are new to me and I haven't concluded my testing yet. The 12ga full and 20ga mod have been shot alot, and have taken quite a few critters. The 20ga gun has taken squirrel at 75 yards with Remington Game loads with 7/8oz of #7 1/2 and #6. It has also taken several feral cats and english sparrows.
The reason I mention the 12ga guns is because I use them with gauge reducers and 20ga shells with great success. The 12ga w/full choke has been my favorite with the above mentioned shells and more so with the Remington Heavy Game or Heavy Dove loads that have a full ounce of #7 1/2 shot. Just for fun, I pattern tested that gun against itself with 12ga and 20ga shells with a 1oz payload of the same size shot. The pattern was the same, which I guess isn't a huge surprise, but of course, I'll take the recoil of a 20ga any day over that of a 12ga if the choice is given.
The 20gauge Heavy Game loads have proven themselves with the 12ga gun. I've been dealing with a major feral cat problem for 6 months. That gun/ammo combination using #7 1/2 birdshot has dropped 15lb tomcats at 25yards through very heavy brush.
There is no doubt that the 20gauge will get the job done. As far as make/model, have you looked into the Remington 1100's? Last year there was a Walmart special-type gun that I wanted to get but didn't. It was a 20-22" barreled youth model, vent rib barrel, screw-in chokes I think, and took 2 3/4" or 3" shells. This year that model was replaced with an 11-87 youth turkey gun. Too bad.