|Personally I would have had a GitHub repo under Minishift org which had just an index file in it keyed by versions which are known to work and can be used. Minishift could then pull down the index file, get the URL for where binary is, which could be anywhere, and then fetch it from there.|
Yes it means the index file has to be updated, but being a separate repo that doesn't need a new Minishift release and could be done at any time.
This would also solve problem of GitHub API rate limiting when Minishift is behind a corporate fire wall, something that I believe was affecting the Minishift developers themselves early on. Not sure if workaround for that was worked out.
So as is always the case, a level of indirection solves all problems.