Any teleconverter/extender made for the Canon EF mount will work with your camera. Your question should be, "Which teleconverter would work with my lenses."
The short answer: none of them.
Both of your lenses are EF-S lenses which will only mount on Canon cameras with crop sensors. The extra tab on the EF-S lenses prevents them from being attached to every converter/extender of which I'm aware in the same way you can't attach your EF-S lenses to a full frame camera.
Even if you modified your lenses so they could fit an EF mount, you probably wouldn't want to do it. Here are the main reasons why:
- You lose maximum aperture and thus auto focus. A 1.4X converter costs you one stop of aperture, a 2X converter costs you two stops. Because your 600D limits your auto focus system to lenses with maximum apertures of f/5.6 or wider, even a 1.4x teleconverter will disable your AF system when using your EF-S 55-250mm f/4-5.6 lens. Adding a 1.4X effectively makes your lens a 77-350mm f/5.6-8 lens. Theoretically you could use AF at 55x1.4=77mm at f/5.6, but once you zoom in past 64x1.4=90mm the max aperture is f/6.3 and past 100x1.4=140mm the max aperture is at f/7.1. As you can see, even before you reach an effective focal length equal to the bare lens' maximum of 250mm you no longer have auto focus capability. A 2X teleconverter narrows the maximum aperture another stop. Your 55-250 f/4-5.6 + 2X becomes a 110-500mm f/8-11 lens.
- You lose image quality. Don't expect the image quality to be as good with the additional glass of a teleconverter between the lens and the camera. In addition to the imperfections added by the additional lens elements, the flaws in the center of your lens will be magnified by the teleconverter. Even the best and most expensive lenses combined with the best extenders will demonstrate some drop in image quality. The lower image quality of your consumer grade lens and the lower optical quality of converters that might work with that lens will give you a much greater hit in terms of image quality. You're going to give up a lot of sharpness, lose a moderate amount of contrast, and increase chromatic aberration. You may also create some irregular geometric distortion as the typical pincushion distortion of the long end a telephoto zoom interacts with the typical barrel distortion created by the teleconverter. While a good bit of the pincushion will be offset by the barrel, what remains will be difficult to correct.
For more, please see this answer and this answer.