Can one conceive of hypothetical physical field for which the energy isn't modeled by the harmonic oscillator but, say, by a infinite square well?
Are all physical fields underpinned by a quadratic potential...?
No. The quadratic-potential case gets lots of attention in textbooks, partly because it's exactly solvable (exactly-solvable examples are rare!), and partly because it's the starting point for a useful approximation method, the method that Feynman diagrams represent.
Why non-quadratic terms are essential
One of the most familiar applications of quantum field theory (QFT) is to scattering experiments. In a scattering experiment, we start with a state of widely separated particles, each of which corresponds to the lowest-energy excitation of some field for the given momentum. The lowest-energy excitation doesn't care about the shape of the potential, as long as the potential admits at least one discrete energy eigenstate above the ground state (otherwise the field would not have any corresponding particle-like excitations). If the potential were quadratic, then "scattering" would be boring: the particles would pass through each other unaffected, completely oblivious to each other's existence. To get interactions between the particles, we need to use a non-harmonic potential — or, more generally, non-quadratic products of two or more different fields. Some intuition behind this is given in J. Murray's answer.
The non-quadratic terms in the Standard Model are what make the Standard Model interesting, and their importance isn't limited to scattering experiments. Life wouldn't be possible without them!
Fermion fields raise another type of exception: the concept of a "potential" doesn't really apply to them, because they are Grassmann-valued fields, but we can still talk about quadratic and non-quadratic terms in the lagrangian, and then the preceding comments still apply.
Can we use a square-well potential?
Consider scalar fields. For any potential $V$ with a finite lower bound, we can construct a quantum field theory of a single scalar field $\phi(x,t)$ whose lagrangian density is
\big(\partial^\mu\phi(x)\big)\big(\partial_\mu\phi(x)\big) - V\big(\phi(x)\big).
If we take $V(\phi)\propto \phi^2$, then we have a free field, which is the harmonic-oscillator case. In that case, the particles don't interact with each other: the theory is boring.
One way to make the theory interesting is to use a non-quadratic potential $V$. For example, the choice $V(\phi)\propto a\phi^2+b\phi^4$ gives what is usually called the "$\phi^4$ model." By tuning the coefficients $a,b$, we can adjust both the single-particle mass and the strength of the interaction between particles. We can also use this model to illustrate spontaneous symmetry breaking.
Yes, we can also take $V(\phi)$ to be a square-well potential, but at sufficiently low energies $V$ might as well be a low-order polynomial, at least if spacetime is four-dimensional. (The story is richer in lower-dimensional spacetime, but I won't go there.) That's because the condition "sufficiently low energy" basically means that only the few lowest-energy modes are excited, even in interactions, and we can tune the coefficients of a low-order polynomial $V$ to reproduce those same lowest-energy modes. For more about this, look up Wilson renormalization.
Using a non-quadratic potential isn't the only way to make the theory interesting, though. Non-linear sigma models use scalar fields that aren't real-valued: they take values in some other manifold instead, such as a circle or some higher-dimensional manifold with non-trivial topology. The topology of the target space (the space in which the fields take their values) makes these theories interesting, even if there is no "potential" at all. The Wikipedia article about chiral perturbation theory introduces an application of this type of model to the low-energy physics of quantum chromodynamics.