In this example, "wouldn't" makes the most sense.
- "Should" pertains to obligation, or what ought to happen
- "Would" pertains to possibility.
Your dialogue is expressing surprise at the possibility that juice may not be healthy, and the expected response would be facts to support that possibility.
Asking why something 'shouldn't' be the case would invite a response containing reasons why something ought not be the case. It can also be used rhetorically - for example, if someone asked "why shouldn't dessert be healthy?", it comes from the viewpoint that desserts are generally not healthy, but and invites anyone to challenge the idea that they could be made healthy.