When a Billing Address is modified, get the new Postal Code. Then check which Contacts on the Account are outside that Postal Code. If 1 or more Contacts are outside of the Postal Code, mark Out_of_Zip as TRUE.

TriggerExample2.tgr

trigger TriggerExample on Account (before update, before insert) {
    Set<Id> changedAccounts = new Set<Id>();
    for(Integer i = 0; i < Trigger.New.size(); i++) {
        Account newAcc = Trigger.New[i];
        Account oldAcc = Trigger.Old[i];
        
        if(newAcc.BillingStreet != oldAcc.BillingStreet ||
          newAcc.BillingCity != oldAcc.BillingCity ||
          newAcc.BillingState != oldAcc.BillingState ||
          newAcc.BillingPostalCode != oldAcc.BillingPostalCode ||
          newAcc.BillingCountry != oldAcc.BillingCountry)
              changedAccounts.add(newAcc.Id);
        
        Map<Id, Integer> accountsAndOutOfZips = new Map<Id, Integer>();
        for(Contact c : [SELECT Id, Account.BillingPostalCode FROM Contact WHERE Id IN :changedAccounts]) {
            if(c.MailingPostalCode != c.Account.BillingPostalCode) {
                if(accountsAndOutOfZips.get(c.Id) == null)
                    accountsAndOutOfZips.put(c.Id, 1);
                else
                    accountsAndOutOfZips.put(c.Id, accountsAndOutOfZips.get(c.Id) + 1);
            }
        }
        
        for(Account acc : Trigger.New) {
            if(accountsAndOutOfZips.get(acc.Id) != null && accountsAndOutOfZips.get(acc.Id) > 1)
                acc.Out_of_Zip__c = true;
            else
                acc.Out_of_Zip__c = false;
        }
    }
}