The per-minute cost for a call to the New Zealand number would probably be a bit on the high side, but not a major problem, and in the worst case you could just ask for the customer's number and call them back. I assume that the CallerID will get lost along the way.
You'll have it up and running in less than a day, you just need to email support and have them enable international calling on your account.
Thanks for explaining this, I actually wrote a blog post (with working sample code) on exactly this topic, because we had some friends in town for Christmas who wanted to call home on Christmas morning without using those really expensive calling cards they were carrying. Basically they called into a U.S. local number with their cell phone, picked from a menu of potential options, and then Twilio dialed to the international number and connected them. Check out my Christmas Eve hack here: http://blog.twilio.com/2009/12/creating-a-custom-internation...
By the way, Twilio rates to New Zealand are 5.2 cents per minute (except for to some mobile lines, which are 47.4 cents per minute - depends on the carrier). http://www.twilio.com/international-calling-rates
I also agree with the article: I personally receive all of our customer calls (which happen like 3-5 times per week). Its a great source of information and it helps our branding that we are a small, customer oriented online retailer.