For a linear system, the superposition principle holds since, be definition, a linear system has the following property:
(1) if $y_1$ is the output for input $x_1$ and
(2) if $y_2$ is the output for input $x_2$ then
(3) the output is $ay_1 + by_2$ for input $ax_1 + bx_2$
In other words, the output for a superposition of inputs is the superposition of the associated outputs.
So, if the differential equation for your system is linear, e.g., the harmonic oscillator, the Superposition Principle holds.
What, then, are you trying to prove?
Prove the superposition principle for inhomogenous linear equations of
motion used in deriving the motion of a driven oscillator. Will it
still apply if the force on an oscillator was −kx2 instead of −kx?
This is, I think, misworded. For example, for the mass on a (linear) spring system, the force on the mass, due to the spring is, by Hooke's law, $-kx$.
A driving force, on the other hand, would be given as a function of time: $F_d = f(t)$.
Then, the net force on the mass is the sum of the driving force and the spring force, $F = f(t) - kx$, which leads to a linear differential equation:
$$m \ddot x +kx = f(t) $$
and thus, the Superposition Principle holds by definition.
This is easy to show by assuming $f(t) = f_1(t) + f_2(t)$ and $x(t) = x_1(t) + x_2(t)$ and inserting into the differential equation.
However, the way I read the problem as stated in your edit, it is the restoring force, not the driving force, that is $-kx^2$.
If that is in fact the case, the resulting differential equation is non-linear
$$m \ddot x +kx^2 = f(t) $$
and thus, the Superposition Principle will not hold since
$$(x_1 + x_2)^2 = x_1^2 + x_2^2 + 2x_1x_2 \ne x_1^2 + x_2^2$$