Here's the review. I haven't read the book, and for all I know the book it might be unpersuasive, but the review did not strike me as particularly apt even focusing only on what's within the review's own four corners.

Consider, for instance, this:

Many of Judge Posner’s arguments in this book are riddled with self-serving contradictions. While he declares that “the Bill of Rights should not be interpreted so broadly that any measure that does not strike the judiciary as a sound response to terrorism is deemed unconstitutional,” he also argues that “a constitutional right should be modified when changed circumstances indicate that the right no longer strikes a sensible balance between competing constitutional values, such as personal liberty and public safety.”

The next example of "self-serving contradiction[]" is this:

In another chapter, which discusses warrantless eavesdropping by the N.S.A., Judge Posner shrugs off the concern that government scrutiny of private communications could lead to embarrassment, intimidation or blackmail of the administration’s opponents. While he acknowledges that “such things have happened in the past,” he says that “they are less likely to happen today” because factors like “the growth of a culture of leaking and whistle-blowing” and “more numerous and competitive media” have converged “to make American government a fishbowl,” and “secrets concerning matters that interest the public cannot be kept for long.”

Later in the book, however, he suggests that people’s privacy (regarding information collected by government data mining) would be better protected if there were more restrictions placed on the news media and “the principle of the Pentagon Papers case” were “relaxed to permit measures to prevent the media from publishing properly classified information.”

Ms. Kakutani goes on to write that "By the end of this chilling book, the reader realizes that Judge Posner is willing to use virtually any argument — logical or not — to redefine constitutionally guaranteed rights like freedom of speech during wartime." The reader naturally expects some examples of illogical arguments of Posner's (since trying to redefine constitutionally guaranteed rights using logical arguments doesn't sound that bad). Here is what the reviewer gives us:

For instance, he expresses irritation with the Supreme Court’s 1969 Brandenburg ruling, which stipulated that speech advocating violence or other criminal conduct cannot constitutionally be suppressed unless it is “directed to inciting or producing imminent lawless action and is likely to incite or produce such action.”

Although Judge Posner writes that “in the present circumstances the enactment of laws forbidding radical Islamist expression would be needlessly provocative,” he ominously adds that “the situation may change” and that he believes “the incitement/threat category could be expanded” to include “generalized advocacy of violence against the United States.”

In his opinion, he says, “to tell Congress and the president that they can do nothing to prevent forms of advocacy likely to multiply the number of future terrorists makes no more sense than telling them that they cannot prevent the publication of recipes for bioweapons because it would probably take years to get from the recipe to the actual manufacture, let alone use, of the weapons.”

Judge Posner believes that “additional counterterrorist measures, in particular in the related areas of electronic surveillance and computerized data mining, could be taken without violating the Constitution (even if there were a clear constitutional right to informational privacy), especially if the effect on privacy is minimized by a strict rule against using information obtained through such means for any purpose other than to protect national security.” And he writes that “coercive interrogation up to and including torture might survive constitutional challenge as long as the fruits of such interrogation were not used in a criminal prosecution.”

Likewise, there's nothing illogical about the arguments for broader surveillance or coercive interrogation. One can respond to these arguments on the merits; but the reviewer doesn't do that — she simply relies on the assertion that these particular arguments are evidence "that Judge Posner is willing to use virtually any argument — logical or not — to redefine constitutionally guaranteed rights like freedom of speech during wartime."

There's more, for instance the reviewer's claim that Judge Posner has "a distinctly cynical outlook that imputes the most mercenary of motives to everyone from journalists to judges: just as Judge Posner has asserted that the media merely pander to the demands of their audiences rather than striving to inform the public, so he suggests in these pages that justices simply 'make up constitutional law as they go along,' following subjective criteria instead of striving to uphold principle and precedent." How is the claim that a judge is deciding based on subjective criteria a claim of "the most mercenary of motives"?

Likewise, consider the claim that "Judge Posner appears to see the Constitution as a fantastically elastic proposition that can be bent for convenience’s sake. 'The greater the potential value of the information sought to be elicited by an interrogation,' he writes, 'the greater should be the amount of coercion deemed permitted by the Constitution. The Constitution contains no explicit prohibition of coercive interrogation, or even of torture, to block such an approach.'"

The trouble is that — especially given a later quote that shows that Judge Posner would bar "the fruits of [coercive] interrogation [from being] used in a criminal prosecution" — Judge Posner's statement that "[t]he Constitution contains no explicit prohibition of coercive interrogation, or even of torture, to block such an approach" is quite correct, and does not at all rely on the Constitution's supposed "fantastic[] elastic[ity]": The Constitution indeed doesn't explicitly prohibit coercive interrogation, so long as the interrogation doesn't involve the person's "be[ing] compelled in any criminal case to be a witness against himself." It may well be that the Constitution should be read as prohibiting coercive interrogation; the Court has certainly so held, and one can read some general concepts, such as "due process" (or for that matter the Ninth Amendment) to include this. But such a reading, even if right, probably requires more "elastic[ity]" than Posner's assertion that "The Constitution contains no explicit prohibition of coercive interrogation, or even of torture, to block such an approach" — an assertion that, if mistaken, is mistaken because it treats the Constitution's text as too rigid, not as too elastic.

So the review is not an impressive piece of work, it seems to me. Again, Judge Posner's arguments may well be unsound, but the review certainly doesn't prove it, or even seem to try hard to prove it (as opposed to using pejoratives to assert it).