What are they saying?

I do not have boxes means I do not have boxes. (1 “not”)

I do not not have boxes means I do have boxes. (2 “not”s)

So, if there is an odd number of “not”s, then I do not have boxes, and if there is an even number, then I do have boxes.

There are 5 “nots”, which is odd, so I must not have boxes.

Let $B$ represent having boxes and $B^{c}$ not having boxes. The above response is logically equivalent to (via DeMorgan's Laws):

$((((B^{c})^{c})^{c})^{c})^{c} \Rightarrow B^{c}$

Hence, the person has NO boxes.

It is reverse philology. Two 'not's mean yes. 4 'not's mean yes. and then 5 'not's mean no.