boxen/boxen may need to know about the github host too.  There are at least a couple places where the host is used in boxen, so I've been working on a PR to address them:

  https://github.com/mdelagralfo/boxen/compare/ghe

First, config.reponame is hard-coded to inspect the origin url for the "github.com" domain; without fixing this, puppet-boxen will fail to find the repo to clone when you first run script/boxen.  Second, boxen periodically uses config.api, which is using the octokit default of the public GitHub API.

In the first case, you could probably parse the domain out of the BOXEN_REPO_URL_TEMPLATE.  The second case is still a bit of a problem if you use the URL template approach.  If the template uses the git protocol, you wouldn't know whether to reference the GitHub Enterprise via HTTP or HTTPS.