In IE, when a .application file is run, the app automatically launches.
On Firefox and other non-IE browsers, the .application file is downloaded. And then the user needs to run that file. This sort of defeats the purpose of having a web launcher - I might as well just ask the user to download an EXE and run it.
Any solutions to this?
I'm not sure about the others, but with FireFox this is a feature. The idea is that they want that one additional layer of "are you sure" before you run any program from the web. So any time it detects that an .exe (or similar) is what is being downloaded, it does not give the option to run from the web, it makes you download the file first.
As @BoltClock said, ClickOnce is a proprietary technology, so it can only be enabled through proprietary addons.
With that said, I've used https://addons.mozilla.org/en-US/firefox/addon/1608/ without any problems. Microsoft also has an officially supported plugin now as well (though I've never used it): https://addons.mozilla.org/en-US/firefox/addon/9449/