In Puerto Rico, during the Settlers phase, are players required to pick a plantation
No, the players are not required to pick a plantation.

From the rules, emphasis mine:


  The player who chooses this role may take either a quarry tile as his privilege or one of the face-up plantation tiles and place it on any empty island space on his player board. Afterwards, each other player, in clockwise order, may take one of the face-up plantation tiles (not a quarry tile! - exception: construction hut) and place it on any empty island space on his player board.