This is transparent if you work in a manifestly supersymmetric formalism such as superspace.
There are two kinds of superfields in a four-dimensional $N=1$ supersymmetric field theory such as the MSSM : chiral superfields and vector superfields. The physical degrees of freedom in a chiral superfield are a fermion, a scalar and a pseudoscalar; whereas in the vector superfield it's a vector field and a fermion. Hence the Higgs fields are component fields of a chiral superfield.
The most general renormalisable N=1 supersymmetric action involving chiral superfields has two kinds of terms:
- a kinetic term, which involves the Berezin integral over all of superspace: $$ \int d^4x d^2\theta d^2\bar\theta \Phi\bar\Phi$$ or, if you abandon renormalisability, then more generally $$ \int d^4x d^2\theta d^2\bar\theta K(\Phi,\bar\Phi)$$ where $K$ is a real function (which can be interpreted geometrically as a Kähler potential); and
- a superpotential term, which is the Berezin integral over "half of the superspace": $$\int d^4x d^2\theta W(\Phi) + c.c.$$
For this to be supersymmetric, the function $W(\Phi)$ must itself be a chiral superfield, which means that it must be a function of $\Phi$ alone and not of $\bar\Phi$. Renormalisability further constrains $W$ to be at most cubic.
So in summary, supersymmetry says that the superpotential has to be a chiral superfield and this in turn means that is a function of $\Phi$, which is tantamount to analyticity.