What is the physical justification for the boundary conditions of the Schrödinger equation for an infinite potential well? All the literature says that the physically meaningful solutions to the Schrödinger equation in an infinite potential well must fulfill the boundary condition that the wave function is $0$ at the walls of the well, otherwise the wave function wouldn't be continuous.
But what differentiates an infinite potential well from a bounded universe with the dimensions of the potential well, where the wave function isn't even defined outside of the "well"/universe? In such a case, the wave function could very well be continuous without being $0$ at the boundary.
What I'm looking for is either an explanation why the case of an infinite potential well is actually different from a bounded universe, or an explanation why the wave function would also have to satisfy the boundary condition in a bounded universe.
 A: There is no a-priori sense in having an infinitely large potential, you can't calculate with infinite numbers (except for some very special mathematical considerations). So, the physical meaning that is given to such a potential must be implicitely contained in the accompanying text that explains the infinite potential well. An this meaning is commonly: "the particle is not allowed to penetrate the outside of the well". Or in mathematical terms: $\psi(x<0)=0$ and $\psi(x>L)=0$.
So you are absolutely right in suspecting that the infinite potential well can be considered equivalent to a bounded universe (with the presumably small dimension of the well, of course), as long as your definition of a bounded universe implies impenetrable boundaries. Note, however, that this is not the only way a (conceptual) universe can be bounded. You could also demand a finite universe with periodic boundary conditions $\psi(0)=\psi(L)$, which would be inequivalent to the infinite potential well.
If you find the boundary conditions of the infinite potential well rather unexpected, think about the similar finite potential well for a moment. This has everywhere "well"-defined potential, so clearly no math crash here. As the closer analysis shows, the walls of the finite well can actually be penetrated. If the energy $E$ of an eigen-state is above the upper potential value $V_{out}$, the solution outside the well is purely oscillatory, as is the case for the inside. But, if the energy is below the upper potential value (so that classically the particle would not be able to be outside), the solution becomes damped oscillatory, with the damping coefficient being proportional to the "missing" energy. So the more energy the particle is missing with respect to the outside potential, the shorter the distance until the wave function has attenuated to $1/e$ or beyond. This is actually the basis of the effect of "quantum tunneling".
Now you can imagine what happens, if you let $V_{out}\to\infty$: the energy of an arbitrary given eigen-state will less and less likely be above $V_{out}$, but more and more likely be below it. So it will be damped more and more strongly. Moreover, the damping goes to infinity because the state is missing more and more energy to the boundary of the well. In the limit, you can consider the wave function as being damped to zero within an infinitely short distance, which is nothing else than what you expect from the infinite potential well. Note, that this is not meant as a rigorous mathematical treatment, but just a mental picture, or as you have desired: a physical justification.
A: 
All the literature says that the physically meaningful solutions to
the Schrödinger equation in an infinite potential well must fulfill
the boundary condition that the wave function is 0 at the walls of the
well, otherwise the wave function wouldn't be continuous.

That's not the reason at all: there are plenty potentials $V(\mathbf{r})$ for particles with 'soft' boundaries that generate continuous wave functions (think quantum harmonic oscillator, e.g.)
For the infinite well, the boundary conditions $\psi(0)=0$ and $\psi(L)=0$ arise from the simple fact that the potentials at these boundaries is $+\infty$ and so no particle can penetrate these potential walls. It's like a prisoner in a cell: he can move inside the cell but he can never escape!

But what differentiates an infinite potential well from a bounded
universe [...]

In a word: scale. Even at very modest scales quantum physics reverts back to Newtonian physics. The Universe is so ginormously large that thinking in terms of bounded wave functions is absurd and a complete waste of time.
