Dead of Winter Exiled Player
Yes, the exiled player still draws a crossroads card and the player on the right still draws one for him/her. 

Since they can't have any survivors in the colony, they can't vote on any cards that require survivors in the colony. 

One of the designers clarified this in a reddit thread https://www.reddit.com/r/boardgames/comments/2kt5fn/dead_of_winter_question_probably_dumb/