Here is the 4th Amendment, which FYI, is part of the Bill of Rights. The "right to privacy" refers to illegal searches and seizures of citizens by the government.
Why, exactly, would a full body scan constitute an illegal search or seizure?
Particularly since the scan is being performed by a private institution to prevent dangerous materials from being smuggled onto their airplanes?
Amendment IV
The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.