Yes, there is a link, both are examples of extremum principles. And yes it is possible to derive the principle of least action from the law of maximum entropy. The derivation is lengthy and I will only sketch the main steps needed:

1. We start from $dS \geq 0$ for an isolated system. For a non-dissipative system, this reduces to $dS = 0$. That is, entropy is a constant.

2. From the phase space structure we can show that the phase space state $\rho$ satisfies the equation $d\rho/dt = \partial\rho/\partial t - \mathcal{L}\rho$, where $\mathcal{L}$ is the Liouvillian.

3. From the constancy of entropy (1), we can derive the Liouville theorem $d\rho/dt=0$, which implies that the general equation of motion (2) reduces to the Liouville equation $\partial\rho/\partial t = \mathcal{L}\rho$.

4. For a mechanical system in a pure state, the phase space state is given by the well-known product of Dirac deltas; substituting this $\rho_\mathrm{pure}$ on the Liouville equation, the equation reduces to the Hamilton equations of motion.

5. Using the Hamilton Jacobi method, the Hamilton equations of motion can be written again as a single equation: the Hamilton Jacobi equation.

6. It can be shown that the [Hamilton Jacobi equation](http://en.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equation#Comparison_with_other_formulations_of_mechanics) "_is an equivalent expression of an integral minimization problem such as Hamilton's principle_", and Hamilton's principle is just the Hamiltonian version of the principle of least action. In other words, solving the Hamilton Jacobi equation one obtains the action $A$ __[*]__ and this automatically satisfies the principle of least action $\delta A=0$.

7. Other versions of the principle of least action can be obtained from here. For instance, the Lagrangian version of the principle can be obtained using a Legendre transformation. 


__[*]__ References on mechanics usually denote the action by $S$, but here it would be confused with entropy.