Yes, Eclipse is more than enough to target all the platforms.
Using PhoneGap Build, you could write the entire app using any text editor - no IDE required - and upload to their service. Their service handles the compilation in all the target mobile environments for you, and then provides the final files to download for each platform.
From their FAQ:
How do I get started with PhoneGap Build?
Simply upload your web assets - a ZIP file of HTML, CSS and
JavaScript, or a single index.html file - to PhoneGap Build, point us
to your Git or SVN repository. Then we’ll undertake the compilation
and packaging for you. In minutes, you’ll receive the download URLs
for all mobile platforms.
For ease of development I'd recommend installing one of the IDEs (probably whatever matches the physical devices you have for testing or prefer). This way you get IDE assistance like Intellisense, but as I said it's not strictly required to have one.
Your alternative if you want to compile for all those platforms locally is pretty ridiculous:
Even if all you have is a web browser you could edit your code in its repository directly through GitHub.com's web interface, and then tell PhoneGap Build to clone the repo and build, then download the builds for each device.
