If I'm wrong, someone please tell me why.
Neither of you are really "wrong". In this case if we took fixie's example of the 10px jpg file and cropped it to 1 px and in width it would be exactly the same as long as we didn't recompress it. However saving a 1px slice straight out of photoshop might end up with banding because of the way photoshop is sampling and encoding the compression of the pixels in the JPG, as fixie pointed out.
This wouldn't happen with a 24 bit PNG and fixie is incorrect in saying browsers render PNGs differently. They are all exactly the same unless we are talking about PNGs with transparency in the case of IE6. Since we're not dealing with transparency in this example, there would be no difference from browser to browser. The 1px wide PNG would also probably look better than a JPG and might even be smaller in file size, depending on the compression level of the JPG.