Can you choose the role with the "New Assignment" event?
You can play it on any player (not just yourself) at any time.  That one player is then allowed to choose a specific role from the available ones.

The thread here has the game's designer clarifying that it is intended to only affect a single player.  He doesn't specifically comment about choosing versus random, but since earlier people mention choosing and he doesn't correct them, it seems this is the correct interpretation.

This makes the card quite powerful, but choosing a random role would be too weak (you're just as likely to get something less suited to your current situation), and allowing everyone to switch roles would be ridiculously overpowered.