How to make a block available on the maintenance page (e.g. jQuery Countdown) ? Is it about setting right permissions?
You cannot. The maintainance page is used for when you have put the site offline, or when the site is experiencing trouble.
In case 1. You don't want your average users to reach the database, after all, no-one knows what state that database is in. The whole point of bringing it down, is to avoid people writing and reading from a crippled, stateless and changing database.
Since Drupal stores its blocks or their configuration in the database, Drupal cannot should not read from the block system.
From the relevant code:
// The bootstrap was not complete. So we are operating in a crippled // environment, we need to bootstrap just enough to allow hook invocations // to work
That said, you can still modify the maintainance template. Just be very carefull with calling Drupal API's and/or the database in there. The maintainance theme is there for the cases when these are not available. And if they are not, your users are going to see ugly fatal errors (or nothing at all).
This answer might come a little late but might be helpful.
What you can do is coding the maintenance page directly from Drupal admin/config/development/maintenance
Here is an example using bootstrap.
Hope this helps you.