Maintenance Page -> Show a specific block
You cannot. The maintainance page is used for when you have put the site offline, or when the site is experiencing trouble. 


When you put the site offline, you typically will be doing database migrations, updates and the likes.
When Drupal puts the site offline, there is trouble; e.g. the databaseserver cannot be reached/is crached etc.


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).