If you have the Daily Quests or Zone Quests for these BGs showing in your Quest Log (L) and those quests show you will receive the box you are referring to, then you will receive them upon turning the quests in to the appropriate NPC listed in the quest. This is tested and confirmed by QA.

If you are referring to the Vanguard Request (H) daily version of the BG quests, please note they do not list a box reward in their rewards. You will only receive the appropriate rewards for completing the Vanguard Request version of these quests as shown in the reward section of the Vanguard Request UI.

If you believe the quest you are completing is indeed intended to deliver a box reward please post a screenshot of the quest you are completing with it's reward section visible so I can correctly reproduce your issue.