In general you can't, because it isn't true. Consider the simple pendulum with boundary conditions $\phi(0)=0$ and $\phi(T)=0$ for some $T$. There are clearly an infinity of trajectories which obey the equations of motion and obey these fixed boundary conditions, including the trivial solution $\phi(t)=\phi'(t)=0$, because the pendulum could go around the loop any integer number of times.
If you carefully follow the derivation of the Euler-Lagrange equations, the statement is that if there exists an extremal trajectory $\phi$ with fixed boundary conditions which extremizes the action $S[\phi]$, then
$$\delta S[\phi] = 0 \iff \frac{d}{dt}\left(\frac{\partial L}{\partial \dot \phi}\right) = \frac{\partial L}{\partial \phi}$$
This does not guarantee the existence of such a trajectory, and if it does exist, it is not guaranteed to be unique. The pendulum problem has the latter feature; for an example of the former, consider the Lagrangian $L(x,\dot x) = \frac{1}{2}\dot x^2 + \frac{1}{2}x^2$ and boundary conditions $x(0)=0$, $x(2\pi)=1$.
The system you describe also features non-unique solutions. In Cartesian coordinates, that's just a particle which is free in the $x$-direction and subject to a harmonic potential in the $y$-direction. If our boundary conditions are chosen correctly, we could have unique solutions or non-unique ones. For example, $x(0)=0,y(0)=0$ and $x(2\pi/\omega)=0,y(2\pi/\omega)=0$ has the trivial solution of a particle sitting at the origin as well as the particle oscillating once in the y-direction.
That being said, for almost all choices of boundary condition, the action-extremizing solutions are unique. In the $x$-direction, fixing $x(0)$ and $x(T)$ fixes the velocity uniquely, because it's a free particle. In the $y$-direction, fixing $T$ (along with the initial conditions) tells you what part of the oscillation you end in; from there, you can show with a bit of algebra that only one choice of initial $y$-velocity will work unless $T=2\pi/\omega$ and $y(0)=y(T)=0$.
In mechanics, however, we often don't talk about trajectories with fixed boundary conditions, but rather trajectories with fixed initial conditions. The uniqueness of these trajectories (and possible conditions for non-uniqueness) follows from the theory of second order ODEs.