Ah, right......they have a slight grey (68,68,68) drop shadow.....ok, no problem......try this...
With the image, (the downloaded icon), open in PS:
CTRL + CLICK the THUMBNAIL of the icons layer....in the 'Layer' panel.
This will place a selection boundary around all non-transparent pixels.
INVERT the selection....Select > Inverse
Hit DELETE a few times...with the MOVE tool selected you can check the Opacity as you do this...hover the cursor over a pixel near the edge of the icon and hit delete until the opacity reads 0%.
This will remove the pixels around the icon.
If the icons DO NOT have to be a specific size for the app to accept them then you could, and I stress could, TRIM the transparent pixels off by using Image > Trim.
If the surrounding pixels get cropped then you know for sure they have all been deleted.
If they DO have to be a certain size then just ignore that and save the image as-is.
If that doesn't work then I dunno, the app obviously supports transparency indicated by the shape of the icons but transparency can be implemented in different ways....like 'matte' for example. if thats the case then finding out the 'matte' colour could be extremely difficult...unless its documented.
Lets see what happens.
Regards.
MrTom.
