I would say the date.
First, I'm not sure why the time zone thing would be a problem in one scenario and not the other. If it's 11:00pm in Los Angeles, it's a different "date" than it is in Boston, but it's also a different number of "days."
But, I digress...
Problem with saying "ships in X days" is, if you're communicating with them over email, they might not be looking at that email until a number of hours have passed. If I buy something today, go to bed and then get the email tomorrow, "X days" now means something other than what I'm interpreting it to mean.
Also, "in X days" can be subjective. Does today count as a day? Does that mean business days? What is a day?
By saying "ships on X date," now you're telling the person EXACTLY when the item is going to ship. No ifs, ands or buts... no room for error or misinterpretation.
That's the way it's done in every carrier(that I know of)'s tracking system. USPS, FedEx, UPS... they all say "This is the DATE it ships, this is the DATE it arrives, etc."
What you could do, if you wanted to be thorough, is to use both:
Ships on December 3, 2016 (2 days from now)