I have both a Kenko 1.4x DG Pro and a Tamron 2x SP 300 Pro.
I got the Tamron 2x because I won a pretty cheap ebay auction. Its image quality is actually pretty good (save for some CA).. Stopping down 1.5 stops is about the same sharpness as wide open without the TC. You do lose some contrast, but you can pull it back in post processing no problem.. The main problem I have with the 2x is autofocus, though.. It likes to hunt a lot, if there's not maaajor contrast with what I'm shooting at. I'll do a lot of manual focus with it, though, no problem. But good luck with fast-moving stuff...
The Kenko 1.4x I got years ago, and it's great. It's a little sharper than the Tamron 2x, and a bittt better with CA, but still has a fair amount of CA.. So you just need to be aware of where/what you're shooting. It's got better contrast, for sure, too. I was more impressed with the 2x that it was so near optically to the 1.4x, after reading all sorts of stuff about 2x complain-complain over the years. I'd say the autofocus is its main problem.
The Kenko 1.4 will do a similar hunting with non f/2.8 lenses, and even with my Sigma 120-300, autofocus speed slows way down. And as someone mentioned, the 1.4x plus the 70-300 IS non-L @ 300 is definitely NOT sharp. Even stopped down, it leaves something to be desired.. But the 70-300@300 isn't sharp in the first place, so you can't expect tooo much. (Really its fine for web-sized photos, though..) Also, it'll hunt like crazy @300, but still try to autofocus.. (essentially it's trying to autofocus at f/8...).
They are both pretty available on eBay, and I've seen them go for anywhere from 50-120-ish... Totally random! Even if you get a 100-400, I would recommend you pick up a 1.4x, if for no other reason than a little bit of fun, and a little bit of "just in case".. It's like an inch thick with end caps.. Definitely worth it having it in the bag, if you ask me! The 2x is significantly larger, and less able to squeeze into the bag unnoticed.. So I'm less inclined to encourage you to go get a 2x.
When I have nothing better to do, I like to stack them, and go crazy.. Interestingly, if the 1.4x is in front of the 2x, I lose 3 stops of light. But if the 2x is in front of the 1.4x, I only lose 2 stops of light.. But in the latter case, the image quality is poorer, and I need to stop down extra, anyway... : D