It sounds like Skyline has having trouble communicating from your computer to our web server.
I do not know why that would be.
You can try downloading the latest MSstats external tool by pressing the "Download MSstats" button on this page:
Then, once you have the .zip file on your computer, you can use the menu item in Skyline:
Tools > External Tools > Add > From File
and point Skyline at that .zip file that you just downloaded.
Let me know if that does not work.