If your pictures are not square to begin with, you cannot change them to a square without distorting the aspect ratio/proportions - they will no longer look right. The 1024 x 1024 refers to the number of pixels; the size in inches would depend on the resolution used. Below is a screenshot of the scaling feature window in Graphic Converter; I don't have Lightroom, but there should be a possibility to scale pics in the file menu somewhere.
In this example, the pic is a standard 8 1/2 x 11; at a 72ppi resolution, that would be 612 pixels wide and 792 pixels long. If I change either of those, the other will change automatically since I have the "proportional" checked.
So, you'll get a square if your pics either are square to begin with or you choose an exact square portion of the image.