Any lens which is compatible with extenders will physically work with any EF or EF-S mount body. However, if the resulting combo has an f8 max aperture (such as a 70-200/4 and 2x TC), it won't AF with most bodies.
The 70-200 II with a 1.4x TC gives you a 98-280/4 lens, which is equivalent to a 157-448/6.4 lens once the 1.6x crop is taken into account.
Likewise, with a 2x TC it's a 140-400/5.6, which is equivalent to a 224-640/9 lens when mounted on the 70D.
As for the quality, take a look at this for the quality on a 60D compared to your old 550D and 55-250 (actually a 50D):http://www.the-digital-picture.com/Reviews/ISO-12233-Sample-Crops.aspx?Lens=687&Camera=736&Sample=0&FLI=4&API=1&LensComp=456&CameraComp=474&SampleComp=0&FLIComp=5&APIComp=1
That's with the 1.4x TC wide open (one stop faster than the 55-250), and the 70D should yield better results than the 60D.
Here's with the 2x TC compared to the native lens, again on the 60D:http://www.the-digital-picture.com/Reviews/ISO-12233-Sample-Crops.aspx?Lens=687&Camera=736&Sample=0&FLI=5&API=2&LensComp=687&CameraComp=736&SampleComp=0&FLIComp=3&APIComp=0
As for the 100-400 II, some people have been waiting years for that. It might be the very next lens to be announced, but equally you could end up waiting another 5 years and see nothing...