This is pretty interesting - the blog post contains a bunch of foreign (non-Japanese) reactions to the phenomenon of mikan boxes in anime (translated into Japanese), and the comments section below contains a bunch of presumably Japanese people reacting to the foreign reactions.
It looks like nobody in Japan is really entirely sure why mikan boxes are so prevalent in anime. However, the following conjectures were made:
- Mikans are the prototypical example of a household product that is purchased in boxes. Since boxes of mikans are frequently available in homes, they are repurposed for things like storage, moving, etc. Thus, people see mikan boxes often. This makes them a good choice for depiction in anime. (comment #30)
- As an extension of this, one commenter (#73) writes that the only boxed product he has in his house is a box of mikans.
- During the Showa period, the mikan box was emblematic of poverty (and in particular, poverty among manga artists in general). Many failed manga artists would live in a tiny apartment and use a mikan box as a desk. Their prevalence in anime is a consequence of the impact they had on artists from that period (or perhaps a sort of self-referential joke by the people in the anime/manga industry). (comments #33, #36, #50, lots of others)
- Mikan boxes are tough and sturdy (as they have to be to prevent the fruit from being smushed in transport), and are easy to come across. They are also available for free at some grocery stores. (comments #43, #47)
- Why mikan as opposed to apple? You can eat a whole box of mikans, but probably not a whole box of apples. (comment #53)
- During election-time, politicians stand on mikan boxes and beer cases and give speeches. (comment #68) [I don't see the connection this commenter is drawing.]
It looks like the most common explanations were 1.) they are one of the few everyday things you buy in boxes; and 2.) as a result of this, many struggling artists, some of whom made it big, spent quite some time drawing using a mikan box as a desk.