Hi Shelly,
Unfortunately, when it comes to long lenses and teleconverters, you get what you pay for and there really aren't any cheap solutions that have any kind of very usable image quality.
As a rule, with 2X you want a 7 element (or more) design to have any hope of decent image quality. But it's going to be extremely dependent upon what lens it's matched up with, too.
Your lens won't autofocus with a 2X on it and the viewfinder will be extremely dark to try to manually focus. A 70-300mm f5.6 will be an effective 140-600 f11. It will also be a lot harder to hand hold steady... I hope you have a tripod or at least a monopod.
In fact, your lens isn't supposed to auto focus with even a 1.4X teleconvter on it (making it an effective 88-420mm f8)... but there are some ways to trick the AF into at least trying to work, when using third party teleconverters on f5.6 lenses.
The Canon teleconverters won't even fit on your lens, so are completely ruled out. All the Canon have a protruding front element design, and that won't allow them to mount on any of your lenses. (Note: the shortest Canon lenses that can be fitted with a Canon TC are the 135/2L and the 70-200mm zooms. Also, Canon calls their TCs "Extenders". That's sometimes confused with macro extension tubes, which are an entirely different animal.)
I also agree with previous response... few zooms work very well with 2X (the newish 70-200/2.8 IS Mark II and Canon 2X Mark III teleconverter are exceptions.... another is the Sigma 120-300/2.8 with a quality TC).
I'd suggest putting the money aside and start saving for a longer lens, such as Sigma 150-500 OS, which sells for roughly $1000 new. It can't really be used with teleconverter, either... But will give you more reach with good IQ for the money.
A bit more expensive alternative would be the Canon 400/5.6L... would likely give better overall image quality than a zoom, but it lacks any kind of stabilization so would largely be a tripod or monopod lens. Some 1.4X teleconverters would be usable with this lens, to give you an effective 560mm f8 lens, though in general you'll lose auto focus (there are some tricks to make AF work to some degree in good light).
For handheld shooting, I use 300/4 IS a lot, both with and without 1.4X (Canon, Mark II... giving me an effective 420mm f5.6). There's a little loss of IQ with the teleconverter... for example the 400/5.6L without any TC would be sharper... but the 300/4 IS and 1.4X is still a very good combo:

When you've got an itch.... black tail mule deer buck
EF 300mm f4 IS lens with EF 1.4X II teleconverter, at f8. EOS 5D Mark II at ISO 1600, 1/250 shutter speed. Handheld, available light (I might have used a monopod, I forget).I bought the 300/4L IS used a couple years ago for a bit under $1000. With current Canon instant rebates, it currently sells new for about the same as the 400/5.6L... roughly $1200. I already had the Canon 1.4X II, but they can be found for around $250 used, maybe a little less (figure $310 new). I have the Canon 2X II, too, but I won't use it with the 300mm (or any of my own zooms). Too much loss of IQ for me.
I would consider the Kenko DG 300 Pro 1.4X teleconverter as a worthy alternative (about $250 new). I would not consider the cheaper Kenko Teleplus MC4 1.4X (about $125).
I have no personal experience with the Tamron, Vivitar and Bower 2X teleconverters you mention. I would bet that the Vivitar and possibly the Bower are actually made by someone else and relabelled. Vivitar doesn't make anything themselves... they outsource everything and commonly just relabel things. I suspect the same is true of Bower, but am not certain. I would suspect, too, that the Vivitar and Bower are the same thing... just with different names.
Tamron, Sigma, Kenko are actual manufacturers... so theirs are unlikely to be outsourced.