Line 15: How many strings do you think you're allocating? Why is Box public?
Line 18: full_list should occur once, not per BoxOfProduce.
Line 19,22: What is the purpose of count? You don't use it.
Lines 33-36: Per the instructions, you should be reading full_list from a text file.
Line 37-46: output() should not populate the BoxOfProduce each time it is called. The instructions say to call output twice. Once after initially populating the BoxOfProduce and again after allowing the user to change the selections.
Line 40: srand() is in the wrong place. It should be called once from main. Not each time output is called. Yes, I know you're only calling output() once, but if you change your program to call output twice per the instructions, you're going to reset the RNG each time you call output resulting in the same sequence of random numbers.
Line 52-57: You don't allow the user to change the contents of his box per the instructions.
PLEASE ALWAYS USE CODE TAGS (the <> formatting button) when posting code.
It makes it easier to read your code and also easier to respond to your post.
http://www.cplusplus.com/articles/jEywvCM9/
Hint: You can edit your post, highlight your code and press the <> formatting button.