The Best Non-Toxic Diapers

Because of limited labeling requirements for diapers, the best solution is choosing a brand that explicitly says their diapers are phthalate free. Of the major brands, Huggies is the best choice, because the company state that their diapers are free of fragrance, phthalates, latex, and parabens. 
Other more expensive brands with a greater commitment to using natural material may have some advantages, but those advantages are more environmental than health-based.  

For example, many bloggers and high-end diaper companies emphasize the importance of Totally Chlorine Free (TCF) diapers, rather than just Elemental Chlorine Free (ECF) (Huggies, for example are only elemental chlorine free) . The suggestion is that even with ECF diapers, using any chlorine in the bleaching processes poses environmental and health hazards, due to the presence of trace amounts of dioxins. The environmental concerns may be valid, but the health concerns are not significant. The possible trace level of dioxins present in the final product is miniscule and orders of magnitude less than we are exposed to in many other ways.

You may need to try a few different brands to find the one with the best fit for your baby, and the best compromise between performance, eco-friendly materials, and cost. Personally, we tried Eco by Naty and a couple of other natural brands before going back to Huggies, because they simply worked better and allowed everyone to get more sleep. Now there are more eco-friendly options available, and if cost was no concern I might consider trying Healthy Baby Diapers, based on their good reviews and EWG verification. 

Of course, cloth diapering is also a great option for avoiding phthalates and other chemicals in disposable diapers. Highly-rated brands include Grovia and Esembly.