You do not need to photograph on a green background to be able to separate them out later. Green screens are used in video so that the process can be automated since many, many images need to be altered (24 to 30 per second of video).

For still images, it is much higher quality to do manual masking to extract the objects since you only need to do it for each photo and you can adjust for issues that often appear when doing chroma keys, such as colored edge highlights (green fringing from backlighting). When you do manual masking, your can get nice white fringing instead that looks much more natural when the image is extracted.

The best results would be to use typical white softbox lighting to make sure that the background caps out at pure white while the product itself is properly exposed. You can then look at pulling any pure white if you want to try automating the process.

If you really want to stick with a green background, there isn't going to be a good way to completely remove the shadow since any light you use to fill the shadow is going to result in producing a shadow of its own. If you do blow out the background however, you can use a light specifically on the area of shadow to blow it out as well, thus eliminating the shadow. If you want to maintain the green, blowing stuff out isn't an option though.