JSON Schema – specify field is required based on value of another field

This is definitely possible with version 3 of the draft. Since you have a complete list of allowed countries, then you could do something like this:

{
    "type": [
        {
            "title": "New Zealand (no postcode)",
            "type": "object",
            "properties": {
                "country": {"enum": ["NZ", "NZL", "NEW ZEALAND"]}
            }
        },
        {
            "title": "Other countries (require postcode)",
            "type": "object",
            "properties": {
                "country": {"enum": [<all the other countries>]},
                "postcode": {"required": true}
            }
        }
    ],
    "properties": {
        "country": {
            "type" : "string",
            "default" : "AUS"
        },
        "postcode": {
            "type" : "string"
        }
    }
}

So you actually define two sub-types for your schema, one for countries that require a postcode, and one for countries that do not.

EDIT – the v4 equivalent is extremely similar. Simply rename the top-level "type" array to "oneOf".