Eric has already given the best answer for this question in the case that the repository does not exist, but I want to point out that:
The response pair of “Repository not found. / remote end hung up unexpectedly” is also given by GitHub when the repository does exist, but the user does not have permission to push to it.
If you’re absolutely certain that the repository already exists on GitHub, make sure the account you’re using has permission to push to that repository.