Which Masters is right for me?
When it comes to picking my masters, I have three options:
  
  M.Sc (Science, thesis based)
  M.A.Sc (Applied Science, thesis based)*
  M.Eng ( Engineering, project based)


The letters in your masters degree do not matter. In other words, you'll never see an M.Sc. or an M.Eng. specifically asked for in an application call ... you'll only ever see "Masters". As such, your degree could be called an M.Xk.Cd. and nobody would really care. It's the topic and quality of your masters matters most! 

If you want to work on quantum computing, then you should look for a good masters programme in either Computer Science, Electrical Engineering or a relevant field of Mathematics. These are typically M.Sc. or M.Eng. courses (again, not that the letters matter!). 

A research masters is usually preferable, in that you may even get a publication or two which will help with PhD applications ... or you may continue in the masters programme and graduate into a PhD programme if you are happy with your supervisor.

On a side note, I would encourage you to be flexible in your interests. You quote "quantum computing" as an interest but it can often be unfeasible to get a PhD you like in a really specific area. I think at this stage it's important for you to decide what discipline you want to pursue (Comp. Sci., Maths, Physics, etc.). Having a sub-field in mind is great, but don't plan your future too rigidly around that.


  A fellow undergrad suggested I do a double-masters to compliment the double-degree, is this possible? Would it be useful to me at all?


Of course it would be useful. :) 

But as to whether it would be worth your time, I would say absolutely not. See (1).

(1) Masters + 0.5 × PhD ≫ 2 × Masters