It depends...PhotoShop and Illustrator can both save as .EPS
If the image was created in Illustrator, then yes, the background can be changed very easily. Select the shape or shapes that make up the background and pick a new color.
As for PhotoShop, an .EPS file is flat (no layers), so the process of changing the background may be a bit more difficult, depending on how complex the image is. Mask out everything in the foreground, or select only the background, by using masks, channels...once the BG is isolated, you can change, adjust or even remove it.
Any background can have its color changed, it just depends on how much effort you want to put in.
Good luck,
Pixel