Butylated Hydroxyanisole aka BHA is as an antioxidant and preservative and its uses include but not limited to:
- food packaging
- animal feed
For humans, various groups and governments are take the following stances:
- Based on animal testing U.S. National Institutes of Health reports BHA is reasonably anticipated to be a human carcinogen
- The International Agency for Research on Cancer classifies BHA as a possible human carcinogen 
- The FDA approves BHA as an acceptable food additive 
- The European Union prohibits the use of BHA as fragrance ingredient in cosmetics
- The State of California, listed it as a carcinogen, and requires warning labels on products containing BHA, notifying consumers that this ingredient may cause cancer 
Is BHA a risk to dogs?
No solid evidence that points to it or anticipates that I can be. Most studies are focused for humans and performed on other animals. Thus, the answer is unclear if it is a risk to dogs or not.
Should you help your dog avoid BHA?
Why not? It is easy to do so. No need to take any risks.
In summary, BHA can potentially do more harm than good for humans. For dogs, the risks are unknown. Most dog food companies avoid using BHA, which gives you a lot of options on the market to select a recipe that does not contain BHA. Why not avoid any potential risk of BHA and check the ingredients to avoid this chemical.
- 13th Report on Carcinogens. U.S. National Institutes of Health, October 2, 2014
- IARC Monographs on the Evaluation of Carcinogenic Risks to Humans vol. 17 (Paris: International Agency for Research on Cancer), vol. 40 (1986).
- “Food additives permitted for direct addition to food for human consumption” FDA Title 21, Volume 3. Revised as of April 1. 2014, CITE: 21CFR172.110.
- “Known Carcinogens and Reproductive Toxicants (California Proposition 65)”. Scorecard. Retrieved 2011-05-29.
- “BHA and BHT” David Suzuki Foundation.
- Shibata, M.A. et al.,”Modification of BHA forestomach carcinogenesis in rats: inhibition by diethylmaleate or indomethacin and enhancement by a retinoid,” Carcinogenesis, (1993 Jul14,7):1265-9.