If I get Earthworm Jim from GOG.com will I be able to use my XBox 360 controller?

I don't use an XBOX 360 controller, but another model. It works just fine for me. It may require some extra tweaking for the game to recognize it, but it should work fine.

I use the Logitech F310 Gamepad. It is essentially designed like a PS3 controller, but the drivers allow it to function with games that are designed for the 360 Windows Controller (it even has the home button) . I managed to get it to work just fine with EWJ, with no additional work. I calibrated it, configured it, and it works beautifully. There is only one caveat. You may be forced to use the analog stick on the Xbox 360. My controller has a button that allows me to force movement to be mapped to the d-pad. To my knowledge, there is no such button for the Xbox 360 controller.

I've just bought the game and I use a 360 controller and I can NOT get it to work properly with the game. In the "Kill All Enemies" screen it's impossible to hit the top two diagonal enemies with the thumb stick and the game has not bound any of the buttons to jump.

Has anyone figured out a fix for this and/or how to bind movement to the D-pad?

The best solution I've found - and it works perfectly - is to use Xpadder, as discussed in THIS thread.

Works fine for me with the 360 pad. Second game has it already set up by just plugging it in and detecting it, but you have to calibrate it on the first game........it is sensitive when setting up so be careful. After that its all smooth sailing.