Sorry, I do not quite understand.
After I replaced my PNG textures to PSD, the problem with a white border for the button disappeared, she was white as needed.
But now there are other problems with transparent textures.
For example, I have a silhouette (attach1 what it looks like in design - SiluetDesign, attach2 my texture in psd - SliuetTexture).
Here's what it looks like when you add in the Sprite (attach3 - SiluetResult).
First, as I understand the level of transparency does not quite match the transparency in Unity3d (Unlit/Transparent Colored shader in atlas texture).
Secondly there is a white border around the texture (attach3).
If I change the Unlit/Transparent Colored shader to Unlit/Premultiplied Colored, all the textures that have transparency are fully (or partially) white. So I set yet on the Unlit/Transparent Colored shader.
I downloaded the filter "Flaming Pear Solidify A" and tried to apply its texture (attach2) but the effect is spread over the entire texture and not just on the border around the silhouette.
I suspect that I did not understand correctly your phrase
...Use PSDs with Alpha instead of transparency
Сurrently my texture in Photoshop (attach2) looks like a transparent background and a certain percentage of opacity of the silhouette.