Is there a way to automatically create a remote repository with EGit if it does not exist yet?
Not that I know of. EGit will use ssh or https in order to connect to the remote server and pass the git command there.
But that won't create the repo on said server anymore than it would if you were to git push to a different folder within your own computer (local computer which you have full control over it as well).
You would get a:
fatal: 'path\to\repo' does not appear to be a git repository
fatal: Could not read from remote repository.