Its impossible to distort a square onto a sphere without having points where it pinches up. There are always going to be some areas which are more distorted than others. I.e. you cant keep squares square when mapped onto a sphere. You can do it to some extent on a torus but I expect that wont help you. All of the map projections have some weaknesses as well as some benefits. You should check out map projections in wikipedia and look for the one which matches the closest to your own ideals. If you then want to warp an image you have using that projection then there are several apps about for it. GDAL is one which springs to mind and there is another with a name similar to GProjector but I forget exactly.

I think the question your asking is if I put some squarish shapes onto the sphere then what will they look like in the original image. Well you can just unproject that image back using the same apps.

Hope that helps.