There’s no problem with solving the Klein-Gordon Equation. It’s just that some of the solutions have a form for which [math]H\Psi=i\frac{d\Psi}{dt}=E\Psi[/math] with arbitrarily large negative values of [math]E[/math]. If we were to try to interpret the solution as a quantum wavefunction this would correspond to having no lower bound on the energy – which is physically unstable.