This achievement requires you to not consume meat, dairy, and alcohol just like the description says. This can be slightly painful, because the game seems to count other items as these three categories that you wouldn't necessarily think are classified as such (such as snacks... they must have meat in them!) Some people have said having survivors eat said items also can glitch the achievement. So if you are planning on saving Snowflake in this same run be sure to use the 2nd method (rather than feeding him steaks) and kill Lenny, then wait until 2pm on Day 1 and head back and find Snowflake auto-tamed. Also be sure not to give survivors alcohol to get the "Survivor drunk achievement".
The EASY way:If you don't mind asking someone for help (like myself), you can ask someone who has a save created right before the military arrives and join their game, spend 5 minutes and get Alpha & Omega AND Purewal Memorial Cup knocked out easy peasy.
Makes playing the campaign a lot more fun when you can eat or drink whatever you feel like!
OTHERWISEThe HARD way:Only drinking ORANGE JUICE seems to be the sure fire bet to not glitching the achievement and there are plenty of locations for it, including the newspapers stand outside of the safe room in Royal Flush Plaza and Roy's Mart in Royal Flush Plaza. Simply always have a stock on hand so that you get forced to use non-orange juice health items in boss fights. Be sure to have at least a couple before each boss fight. Never carry any other food items so that you don't accidentally use them in a panic. Grab the magazines for health boosts. One in the Ragazines shop in Palisades Mall and one in the Newspaper Stand in Royal Flush Plaza outside of the safehouse. Once you've made it through the whole story without failing any of the Case files and you end up on the Royal Flush Plaza rooftop when the military arrives, the achievement will unlock.