I use Windows and I don't think I have an icon corresponding to the 'eyeball' you mention.
I can do a right-click and choose Open, or 'Open With' followed by FoxIt from the context menu - this does work. Less convenient than a double-click, though.
Since I initiated this thread I have seen what looks to me like the same PDF interface in other software products and on the web. So I think that it must be a software widget.
However, it seems that very little thought has been given to interfacing it conveniently with Evernote and there are bugs - for example, regarding pasting a number of PDFs into a note as a single operation. This doesn't always result in all of them being pasted. Further, the order in which they appear is strange and is not what it used to be!
The other big issue for me is that it has changed my whole way of working with Evernote to the extent that I may not ever have chosen it as my software tool. Or I may have done things differently. As it is, over the course of several years I have built up a database and a way of working that has now been crippled.