For example, are you asking what the volume of this solid is?
Imagine a very thin slice of this solid, perpendicular to the x-axis. It is, as you say, a square and let's say it's thickness is dx. According to you, one side of the square runs from -sqrt(1- x^2) to sqrt(1- x^2) and so is 2sqrt(1- x^2) long. What is the AREA of that square? The volume of that thin slice is that area times dx. Integrate that to find the volume. The limits of integration are, of course, the x values where the two semi-circles come together.