I use the 300/4 IS with a 1.4X TC.... That gives me two really useful focal lengths that are easily handheld.
This was shot with the 300/4 on full frame 5DII, so you'd actually see more "reach" using it on your 60D (1.6X)...
This was done with 300/4 plus 1.4X (Canon, Mark II) on 5DII...
This is the 300/4 alone on 7D (same crop as your 60D, of course)....
And this is the 300/4 plus 1.4X on 7D....
One nuance of this lens is that it tends to have a little magenta cast in strong highlights. If a problem, it's easily fixed in post processing. IMHO, it wasn't a problem so I didn't fix it in the catchlight in the eye of the hawk, above...
The zooms will be more versatile or convenient and offer just slightly less reach than the 300/4 + 1.4X. I've been looking at the 120-400 myself, to use for sports in reasonably good lighting situations (until the Canon 200-400/4 1.4X is finally released... and I win the lottery). The Canon 100-400 doesn't really appeal to me, I'm just not a big fan of push/pull zooms (other folks really like them, so this is just me).
There are also the Sigma 150-500 and 50-500.... Both are larger, heavier. The 150-500 is only a little more expensive than the 120-400. The 50-500 usually sells for about the same as the Canon 100-400. (I notice that you are in Sweden and the prices seem quite different there)
The ideal might be Sigma 120-300/2.8 OS with a quality 2X teleconverter. That would cover a lot of bases in a single lens! Of course, it's double the price of the 100-400 or 50-500. It's also the largest and heaviest of this bunch.
There is no such thing as "too long" a telephoto for birding. If you have a 400mm, at times you'll wish you had 500mm. If you have 500mm, there will be times you wish you had a 600mm... 800mm, etc.
Sigma is in the process of selling out the current version of the 120-400 OS, 120-300/2.8 OS and possibly some of the other lenses mentioned... They are revising their entire lens line right now. AFAIK, the main difference will be the new lenses can be user programmed to fine tune focus performance and will have a new finish. Optical formulas, AF system, and OS performance don't seem to be changing so image quality might not be effected at all.