I see that the download path for a GitHub repo is of the form
https://github.com/{username}/{reponame}/archive/{branchname}.zip
For a priva
Here is a pure C# way:
var githubToken = "[token]";
var url = "https://github.com/[username]/[repository]/archive/[sha1|tag].zip";
var path = @"[local path]";
using (var client = new System.Net.Http.HttpClient())
{
var credentials = string.Format(System.Globalization.CultureInfo.InvariantCulture, "{0}:", githubToken);
credentials = Convert.ToBase64String(System.Text.Encoding.ASCII.GetBytes(credentials));
client.DefaultRequestHeaders.Authorization = new System.Net.Http.Headers.AuthenticationHeaderValue("Basic", credentials);
var contents = client.GetByteArrayAsync(url).Result;
System.IO.File.WriteAllBytes(path, contents);
}