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The reason a .22 is preferable to a shotgun as an anti-zombie weapon is primarily ammunition weight.

Since the only way to bring down (Brooksian) zombies is to destroy the brain, all rounds capable of penetrating the skull are equivalent. As long as your chosen .22 round can penetrate the skull you don't gain any advantage by increasing you firepower to that of a shotgun, and you can carry far more rounds of .22lr than 12 gauge.

I do disagree that a melee weapon should be your primary weapon. Since zombieism is transmitted by bites you want to avoid close combat whenever possible. A long gun should be the primary weapon (an AR15 is ideal for this, with good long-range accuracy and relatively light ammunition), a pistol (probably a .22) the secondary weapon, and a melee weapon (preferably a tool you would already be carrying) as one of your last options.