What is the significance behind the apples and boxes?
I haven't seen it myself, but apples can often symbolize sexuality, temptation ("forbidden fruit") , fertility ("bearing fruit" as in your screenshot). 

Connections can also be made to the stories of Adam &amp; Eve and Sleeping Beauty.

The circular shape of an apple can also be a depiction of loops, or eternity - which is what it appears to be in Penguindrum:


   It is revealed in the finale that the apple is a metaphor for the Penguindrum, which seems to be Kanba's life, shared between the three siblings


(TVTropes)

This article is more indepth about the apples in Penguindrum