Born rule and unitary evolution
Strictly speaking, the Born rule cannot be derived from unitary evolution, furthermore, in some sense the Born rule and unitary evolution are mutually contradictory, as, in general, a definite outcome of measurement is impossible under unitary evolution - no measurement is ever final, as unitary evolution cannot produce irreversibility or turn a pure state into a mixture. However, in some cases, the Born rule can be derived from unitary evolution as an approximate result - see, e.g., the following outstanding work: http://arxiv.org/abs/1107.2138 (accepted for publication in Physics Reports). The authors show (based on a rigorously solvable model of measurements) that irreversibility of measurement process can emerge in the same way as irreversibility in statistical physics - the recurrence times become very long, infinite for all practical purposes, when the apparatus contains a very large number of particles. However, for a finite number of particles there are some violations of the Born rule (see, e.g., the above-mentioned work, p. 115).
The use of the word "postulate" in the question may indicate an unexamined assumption that we must or should discuss this sort of thing using an imitation of the axiomatic approach to mathematics -- a style of physics that can be done well or badly and that dates back to the faux-Euclidean presentation of the Principia. If we make that choice, then in my opinion Luboš Motl's comment says all that needs to be said. (Gleason's theorem and quantum Bayesianism (Caves 2001) might also be worth looking at.) However, the pseudo-axiomatic approach has limitations. For one thing, it's almost always too unwieldy to be usable for more than toy theories. (One of the only exceptions I know of is Fleuriot 2001.) Also, although mathematicians are happy to work with undefined primitive terms (as in Hilbert's saying about tables, chairs, and beer mugs), in physics, terms like "force" or "measurement" can have preexisting informal or operational definitions, so treating them as primitive notions can in fact be a kind of intellectual sloppiness that's masked by the superficial appearance of mathematical rigor.
So what can physical arguments say about the Born rule?
The Born rule refers to measurements and probability, both of which may be impossible to define rigorously. But our notion of probability always involves normalization. This suggests that we should only expect the Born rule to apply in the context of nonrelativistic quantum mechanics, where there is no particle annihilation or creation. Sure enough, the Schrödinger equation, which is nonrelativistic, conserves probability as defined by the Born rule, but the Klein-Gordon equation, which is relativistic, doesn't.
This also gives one justification for why the Born rule can't involve some other even power of the wavefunction -- probability wouldn't be conserved by the Schrödinger equation. Aaronson 2004 gives some other examples of things that go wrong if you try to change the Born rule by using an exponent other than 2.
The OP asks whether the Born rule follows from unitarity. It doesn't, since unitarity holds for both the Schrödinger equation and the Klein-Gordon equation, but the Born rule is valid only for the former.
Although photons are inherently relativistic, there are many situations, such as two-source interference, in which there is no photon creation or annihilation, and in such a situation we also expect to have normalized probabilities and to be able to use "particle talk" (Halvorson 2001). This is nice because for photons, unlike electrons, we have a classical field theory to compare with, so we can invoke the correspondence principle. For two-source interference, clearly the only way to recover the classical limit at large particle numbers is if the square of the "wavefunction" ($\mathbf{E}$ and $\mathbf{B}$ fields) is proportional to probability. (There is a huge literature on this topic of the photon "wavefunction". See Birula 2005 for a review. My only point here is to give a physical plausibility argument. Basically, the most naive version of this approach works fine if the wave is monochromatic and if your detector intercepts a part of the wave that's small enough to look like a plane wave.) Since the Born rule has to hold for the electromagnetic "wavefunction," and electromagnetic waves can interact with matter, it clearly has to hold for material particles as well, or else we wouldn't have a consistent notion of the probability that a photon "is" in a certain place and the probability that the photon would be detected in that place by a material detector.
The Born rule says that probability doesn't depend on the phase of an electron's complex wavefunction $\Psi$. We could ask why the Born rule couldn't depend on some real-valued function such as $\operatorname{\arg} \Psi$ or $\mathfrak{Re} \Psi$. There is a good physical reason for this. There is an uncertainty relation between phase $\phi$ and particle number $n$ (Carruthers 1968). For fermions, the uncertainty in $n$ in a given state is always small, so the uncertainty in phase is very large. This means that the phase of the electron wavefunction can't be observable (Peierls 1979).
I've seen the view expressed that the many-worlds interpretation (MWI) is unable to explain the Born rule, and that this is a problem for MWI. I disagree, since none of the arguments above depended in any way on the choice of an interpretation of quantum mechanics. In the Copenhagen interpretation (CI), the Born rule typically appears as a postulate, which refers to the undefined primitive notion of "measurement;" I don't consider this an explanation. We often visualize the MWI in terms of a bifurcation of the universe at the moment when a "measurement" takes place, but this discontinuity is really just a cartoon picture of the smooth process by which quantum-mechanical correlations spread out into the universe. In general, interpretations of quantum mechanics are explanations of the psychological experience of doing quantum-mechanical experiments. Since they're psychological explanations, not physical ones, we shouldn't expect them to explain a physical fact like the Born rule.
Aaronson, "Is Quantum Mechanics An Island In Theoryspace?," http://arxiv.org/abs/quant-ph/0401062
Bialynicki-Birula, "Photon wave function", 2005, http://arxiv.org/abs/quant-ph/0508202
Carruthers and Nieto, "Phase and Angle Variables in Quantum Mechanics", Rev Mod Phys 40 (1968) 411; copy available at http://www.scribd.com/doc/147614679/Phase-and-Angle-Variables-in-Quantum-Mechanics (may be illegal, or may fall under fair use, depending on your interpretation of your country's laws)
Caves, Fuchs, and Schack, "Quantum probabilities as Bayesian probabilities", 2001, http://arxiv.org/abs/quant-ph/0106133; see also Scientific American, June 2013
Fleuriot, A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton's Principia, Springer, 2001
Halvorson and Clifton, "No place for particles in relativistic quantum theories?", 2001, http://philsci-archive.pitt.edu/195/
Peierls, Surprises in Theoretical Physics, section 1.3
This is clearly a somewhat controversial topic, but Zurek claimed to derive the Born Rule from other postulates.