In general, the Halacha permits engaging a non-Jew in a task, as long as it was not stipulated that the work be performed specifically on Shabbat and the payment was a fixed price. This is the basis for permitting sending of mail or submitting clothes to the cleaners on Friday. Even though the non-Jew may do the task on Shabbat, it is considered as though he is working for himself. It was his choice to do so for his own convenience. As far as the Jew is concerned, the task could have been performed after Shabbat.
However, sending express mail on Friday poses a serious Halachic problem. Since it "absolutely positively has to get there overnight," it is tantamount to the Jew directly instructing the non-Jew to work on Shabbat. There is no way for the letter to arrive at the specified time without non-Jews transporting it on Shabbat at the behest of the Jew.
The Shevut Yaakov, in his responsa, discusses a possible leniency to send express mail over Shabbat. He bases his approach on a famous dispute between Rabeinu Gershon and the Havot Yair as to whether Amira L’Amira is permissible on Shabbat. That is, is a Jew allowed to instruct a non-Jew to instruct another non-Jew to perform melacha on Shabbat? While telling a non-Jew to work is certainly prohibited, does the addition of an intermediary render it permissible? According to the Havot Yair (siman 49), Amira L’Amira is permitted for the sake of a misva or in the event of monetary loss. Therefore, if one submits an envelope to a non-Jewish employee or postal clerk who then gives it to the non-Jewish mail carrier, e.g. Staples or a mailbox company gserving as agents to give the letter to UPS, this would constitute Amira L’Amira. Such a case would be permitted for the sake of misva or other urgent need.
However, one should be careful not to pre-label the package himself with the address and delivery instructions. This may constitute a directive from the Jew directly to all of the non-Jews engaged in the delivery.
This approach is adopted by the Shemirat Shabbat K’hilchata (Ch. 31), Rav Pinhassi in his work V’Daber Davar and Hacham David in his work on Amira L’Akum.
It is permitted to send express mail over Shabbat only for the sake of misva or urgent need, as long as the Jew does not pre-label the package with delivery instructions.