The file format .gif uses color palettes, and does not support general alpha channels. But you can select an individual color and specify that color should be interpreted as transparent. Hence, in your case, you want white:
Export["mygif.gif", mygif, "TransparentColor" -> White]
Here's one frame as interpreted by Photoshop (where transparency is shown by the grey boxes)
In response to the query about making the eyes transparent, this can be done by quantizing the color. For example:
Export["mygif.gif", ColorQuantize[gif[], 8],
"TransparentColor" -> RGBColor[228/255, 34/255, 15/255]]
quantizes the bulk of the eyes to a color near red, and then chooses that color to be transparent. Of course, when you do this, the background it no longer transparent. To do that, you could change the reddish pixels to white, and then use
"TransparentColor" -> White.