Others have mentioned my main arguments against this line of reasoning, so I'll just do a quick implied proof by contradiction to show that sacrificing privacy for security is useless to begin with.
Prisoners in the US have essentially zero privacy, and drugs AND weapons still manage to get into prisons. That is to say, prisons, the place with the least amount of privacy, are not even twice as "secure" as the public. Not only that, murders, rapes, and beatings, by guards and other prisoners, go unpunished most(?) of the time, especially when committed by those in power (the guards) without any unbiased oversight (that is, oversight that doesn't solely consist of guard/police unions or those on their payroll).
If getting rid of 99% of your privacy doesn't even double your security, why volunteer, well, ANY of your privacy to a government? Sure, maybe sacrificing 5% privacy might yield a +10% security, and additional privacy sacrifices give diminishing returns, but why trust an obviously corrupt government with not abusing your waived private information?
edit:
UrKnightErrant said:
Under American law freedom is endowed by the laws of nature and of nature's God, not government, and extends to "all men" (not just Americans). It is a birthright, not a privilege extended by a benevolent bureaucracy or majority.
I fixed a small bit, with some emphasis, using the wording from the Declaration. The distinction, I feel, is pretty important, since many of the Founding Fathers, particularly the primary author of the Declaration, were deistic in belief, believing that the god that created the universe didn't play an active part in it beyond creation; that we live in a world of men, and that men decided man's fate; no miracles or divine intervention or the like. Our rights are natural, but not exactly "endowed" by a god. The way the latter statement is worded makes our country appear far more fanatical than the Nation of Men we were meant to be. "Nature's God" is a specifically deistic concept in the way that one might say "Mother Earth" or "Father Time," and isn't really referencing a 'real' deity like Yahweh.
At the very least, the Declaration of Independence ties the Laws of Nature into Nature's God, such that either or both could be the explanation for our rights, or that they are one in the same. It's hardly a "god-given" right (as many Americans seem to think these days) as worded originally.