What is browserstack drive? Github and bitbuckets are git repositories. Github will let you host public repository and you can make your images public. In case of bitbucket, this will be little challenging as you will need credentials to work with bitbucket repository. I would suggest you to get one http or ftp server and put your files there.
If you are using Github public repository, make sure that you get permission from the author. As anyone accessing Github will be able to access your file easily.
Hope that answered your question.