Is boxed water really better? Better than what? Strangely enough, Boxed Water doesn't provide information on whether they conducted a Life Cycle Assessment to back-up their claim. 

It looks like a big difference with plastic bottles is about shipping, according to this article. For one truck's worth of bottled water, Boxed Water can deliver 26 trucks' worth of cartoned water. 

Furthermore, the manufacturing of boxed water is estimated to emit about 8 grams of greenhouse gases (GHG) emissions where a PET plastic bottle is responsible for about 50 grams of GHGs. 

Another advantage is that the entire carton is recyclable. See how it is done here. However according to this other article, only 52% of consumers leave in communities that fully recycle the cartons. But even if the box is not recycled, it can be flattened and will use less space in a landfill than a plastic bottle. Also because about 75% of the boxed water is made out of paper, it can decompose, unlike plastic. There is now an even better option with Just Water, the plant based  boxed water. 

So boxed water is probably better than plastic bottles but refillable bottles should still be the best option.