There is not a direct link between the linearity of some physical laws and the superposition of quantum mechanics. The latter is more of a special kind of linear superposition which requires some restrictions on the coefficients.
The existence of the phenomenon of superposition of states is a characteristic of quantum mechanics. In classical mechanics such a phenomenon does not exist because every irreducible representation of the corresponding algebra of observables is one dimensional. In order to have a superposition of states, higher dimensional irreducible representations are necessary, where the state space can then be identified with the projective representation vector (Hilbert) space. When at least two independent unit vectors, say $u,v$, are then available, one can construct any superposition
$$w = \alpha u + \beta v,\qquad\alpha,\beta\in\mathbb C,$$
with the condition that $|\alpha|^2+|\beta|^2 = 1$. This is necessary to ensure that $w$ defines a state, i.e. a normalised linear functional on the algebra of observables. When you look at the operation of superimposing the states generated by $u$ and $v$ into the state generated by their superposition $w$ you see that this map is not linear, for once because the projective Hilbert space (which is in a one-to-one correspondence with the accessible states) is not a linear space (think of this as a map that takes two points on a sphere and spits out another point on the sphere; this analogy is not entirely perfect in this case but close enough, modulo some identifications of points).
When there is a equation that governs the dynamics in this framework, then the superposition principle applies if this equation is linear. The meaning of superposition is a bit different in this context. While you can still superimpose states in the sense above, if $u$ and $v$ are solutions of a non-linear equation, then $w$ need not be a solution of the same non-linear equation; nonetheless it is still a valid superimposition of the states $u$ and $v$.
