The answer is multifaceted. Let me first observe that Dr. Peterson is making a substantive and observable point. For example, Sam Harris is quoted as saying the following at the Beyond Belief conference in 2006:
"It is rather more noble to help people purely out of concern for their suffering than it is to help them because you think the Creator of the Universe wants you to do it, or will reward you for doing it, or will punish you for not doing it. The problem with this linkage between religion and morality is that it gives people bad reasons to help other human beings when good reasons are available."
As we can see above, what the New Atheists have done is taken a religious principle (in this case Christian altruism) and merely removed it from God. Often in response to such accusations one might argue, as did Christopher Hitchens, that these ideas are not based in Christianity but can be found in a variety of societies and cultures over time. This is why I would argue that Dr. Peterson is wrong to call these ideas as requiring "Christian" metaphysical presuppositions. It is more proper to consider them "religious" metaphysical presuppositions since they are not limited to Christian constructs.
Why does this matter? It has nothing to do with intellectual dishonesty: the New Atheists seemed to have shared values beyond the dismissal of God which include personal values like altruism, helping the poor, and a love of a Truth they argue is "good" inherently. However, it is more something like this: atheists have spent years debating theists like Frank Turek who, in his famously frustrated debate with Christopher Hitchens, question where athesit morality comes from. The New Atheists have taken the same route many other atheists have traveled: they endeavor to show that a person can be "good" without God, or more specifically that a person can be just as good as a "good Christian" without God.
What they have done is to demonstrate that one can live up to the Golden Rule, or to the few parts of the Commandments they like, or have proof of Truth-as-good without God. It is like trying to live up to the standards of your admittedly abusive parents while dismissing them for their cruel behavior. The New Atheists are saying, in effect, "Look at me! I am a good Christian, but without the need for supervision." With Hitchens as an exception (who seemed to march to the beat of a different drum on a great majority of these issues), the New Atheists as a whole seem willing to prove how they can be good Christians without a God.
But, if asked what it means to be good, they will accept some sort of religious ethic, often not proudly but as an exhausted, last ditch option. They say things like: help the poor, because it is better to reduce suffering in the short time you have than prolong it. It is better to do unto others as you want done, because life is short and we suffer because of it. They accept the ethic, to whatever degree, as the ultimate sign against the God they despise, hoping to get some sort of praise for being so "good" (in a borrowed sense, obviously) without God.
I make this point for the sake of differentiating the New Atheists from the postmodern or Meta-modern Atheist. This Atheist is at first glance much like the New Atheists, tearing down the obvious contradictions of religion in a literal sense over and over again. They show you just how much a literal reading of the Bible, the Quran, the Torah, and so on makes no sense regarding any rational sense. But the Meta-modern Atheist, however, has come to a very complicated, and perhaps dark, conclusion that is different than the premise of the New Atheists. The true or proper Atheist, according to the Meta-modern Atheist, is one who completely pulls themselves away from religion. They recognize that one can only do so by rejecting even the idea of being good by the standards of God or the religious. You must stand tall, in the face of all things, and say, "I am good, not by your standards, but by my own standards."
To become a meta-modern atheist you must ask yourself, without giving into guilt; "What does it mean to be good?" "How do I know?" And once this has been found out, independently, you can start decided what values you accept. We are not here to dress up for the religious, to say, "Look at me, I give alms just as well as you, without God." The point is to say, "I give alms because of my standards, because my reasons are better than yours." Often, I would argue that the Meta-modern Atheist may not even need to mention God; for God truly to the Meta-modern Atheist does not exist outside of a historical concept. Meta-modern Atheists figure out what is good and what it means to be ethical, instead of surrendering, however loudly, to religious dictations of such terms.
Atheism, as a meta-modern endeavor, is not a fight, my friends. Meta-modern Atheism is the first step in developing who you are. It is the first step in the project of your own humanity. Find your own ethics, don't borrow it. Reconstruct your beliefs, don't deconstruct.