For the last three weeks, when I do the Blackguard Defenders quests, I have been using Protector Yi as my companion. I have used him four or five times to the end of the quest chain, but he still has not been checked off for the achievement; since you are supposed to do 15 quests with him, and there are 5 quests for day, that should mean that you only need to pick him three times if you keep him with you. I do, yet he is still not checked off.
Today, when I picked up the missions, I also took the Monk companion, Lao-Chin the Iron Belly. However, as soon as I picked him up, I dismissed him in order to try for the solo achievement. I did all the quests, returning to the Shado-Pan camp to pick up the last quest. When I turned that in as well, I did not get the achievement for doing the quests solo.