关键在于商家是否有“有意”出售过期食品

it is quite hard for her to persuit that case because she will have to prove that:

the food market is intentional in deceiving customer into buying out-dated food.

and also she will have to show that her son's diarrhea is a result of drinking that juice and not from something else.

both of which are pretty hard to prove, therefore it's not a good case.