Due to poorly made assumptions in the script and the lack of error checking, the ~/bin/particle you downloaded is an empty file, marked with executable permissions.
As a workaround, I’d recommend downloading the amd64 (Intel Mac) version of particle directly.
curl -fsLo ~/bin/particle --create-dirs \
https://binaries.particle.io/cli/master/1.1.0-12d6799/darwin/amd64/particle
chmod +x ~/bin/particle
Afterwards you should be able to log in and install additional dependencies:
particle login
brew install dfu-util
brew install openssl