How can I claim ownership of my idea without securing a patent?
You can't patent that.


  Can I patent the use of an existing compound for another purpose since I have reason to believe it would also act to target neurodegeneration in this disease?


No, you can't patent "the use of X for purposes of Y" abstractly. As this description of patentable matter explains it:


  A patent cannot protect an idea. Instead, the idea must be embodied in one or more of the following:
  
  
  A process or method (such as a new way to manufacture concrete)
  A machine (something with moving parts or circuitry)
  A manufactured article (such as a tool or another object that accomplishes a result with few or no moving parts, such as a pencil)
  A new composition (such as a new pharmaceutical)
  An asexually reproduced and new variety of plant.
  


If you developed a specific method to use compound X to do neurorestoration/neuroprotection of Y - then maybe you can patent it.

Instead of trying to patent - try scientific publication!

You need the anxiety to "capitalize on your effort". Like you, I have not been able to secure a position in academia - not even a temporary one - despite having a bunch of solid ideas which can be developed. So I can sympathize, at least somewhat. Still: You should not, must not, utilize the tool of patents - which is a legal mechanism for inhibiting scientific discovery and preventing the public from benefiting from inventions.

Instead, try to find the time to write a paper - albeit a "vision paper" rather than an empirical study - making your case. Perhaps publish it on arXiv before it's peer-reviewed, to "stake your claim" scientifically rather than capitalistically. It might help you find a source of funding for developing this idea - in a "start-up", an established company or even a scientific institution.