It's not clear why packing boxes would require security clearance. Unless, of course, the issue has less to do with sensitive government secrets than with the ever-expanding U.S. security clearance system.
As many as 4 million people hold "top secret" security clearance, of which 500,000 are private contractors. One reason for this trend is that the U.S. government has become so reflexive about classifying information, much of which is not nearly as sensitive as an NSA spying program, that clearance are required even for totally banal work.
One effect of this classification of nearly everything, and subsequent granting of clearances to nearly everyone, is that all it takes is one or two loose cannons among those 4 million clearance-holders to spill out government secrets. Whether or not you think Edward Snowden was morally right to release the information about U.S. telecommunications monitoring to the public, he represents what is becoming a significant problem for U.S. intelligence.
As the Washington Post's Greg Miller put it, Snowden and Bradley Manning, who released thousands of classified diplomatic cables to WikiLeaks, are both "leakers who worked at the lowest levels of the nation’s intelligence ranks but gained access to large caches of classified material."
The intelligence community's weakness here is that its classification of so much data and its reliance on private contractors mean that lots and lots of people who might not need access to top secret data are still required to get the clearance. And once there are millions of people poking around behind the clearance wall, it's only a matter of time until one of them decides to release some of it. Because once you're granting so many people access to top secret information that even professional box-packers are on the list, it's not really so secret anymore.