For the 2-inch squares, starching the fabric (as heavily as you can) before cutting would be helpful. Starch will stabilize the fabric so the edges don't stretch. Ideally you also want to starch the block so the block edges don't stretch as you attach the squares.
A little bit of fullness will generally "quilt out". As long as you're not actually sewing puckers and tucks into the top when you quilt, chances are it will be just fine.
If you are machine quilting, I recommend heavily starching the background fabric before layering, as this also prevents tucks and puckers in the backing.