As a first pass, you could assume that the balloon material is inextensible (i.e., that it can't stretch) and has no bending stiffness. Then the resistance to squashing a balloon arises solely from the pressurization of the internal gas, which we'll assume to be ideal.
Assume that the compression makes the balloon look like a sphere that's lost a spherical cap:
The remaining volume $V(h)$ is then
$$V(h)=\frac{4}{3}\pi r^3-\frac{\pi h^2}{3}(3r-h). $$
The resistance force $F$ when you compress a system arises from the increase in energy $U$ per unit compression $h$:
$$F=\left(\frac{\partial U}{\partial h}\right)_S.$$
Here, I've written this statement in partial differential form; don't worry if you haven't covered this material yet. The $S$ at the lower right stands for constant entropy, which simply means that the squashing occurs quickly (as opposed to very slowly, where we might assume that instead it's the gas temperature that stays constant).
I can apply the chain rule of differentiation to expand this term into things I know:
$$F=\left(\frac{\partial U}{\partial h}\right)_S=\left(\frac{\partial U}{\partial V}\right)_S\left(\frac{\partial V}{\partial h}\right)_S=-P\left(\frac{\partial V}{\partial h}\right)_S,$$
where $P$ is the pressure. (We know from the fundamental relation that $(\partial U/\partial V)_S=-P$ for simple closed systems.) It turns out that for an ideal gas being compressed quickly, $$PV^\gamma=P_0V_0^\gamma,$$ where $P_0$ and $V_0$ are the initial pressure and volume before compression and $\gamma$ is the heat capacity ratio, which can be approximated by 1.4 for air, for example. Thus,
$$F=-P_0\left(\frac{V_0}{V}\right)^\gamma\left(\frac{\partial V}{\partial h}\right)_S.$$
Note that $V_0$ is simply $\frac{4}{3}\pi r^3$. We can determine $\left(\frac{\partial V}{\partial h}\right)_S=\frac{dV(h)}{dh}$ from differentiating the volume expression above:
$$\frac{dV(h)}{dh}=\pi h^2-2\pi hr,$$
if I differentiated correctly. The final unsimplified expression is
$$F=-P_0\left(\frac{\frac{4}{3}\pi r^3}{\frac{4}{3}\pi r^3-\frac{\pi h^2}{3}(3r-h)}\right)^\gamma\left(2\pi hr-\pi h^2\right).$$
Try simplifying this and plugging in some expected values. Do the results make sense?
