= Solution
A Klein-Gordon solution is second order in time: both $\Psi$ and $\partial_t\Psi$ are needed as <initial data>. Its free modes have both signs of <frequency>,
$$
E=\pm\sqrt{c^2p^2+m^2c^4}.
$$
This differs from the first-order Schrödinger evolution of a fixed one-particle <wavefunction>.
Moreover, $|\Psi|^2$ does not generally obey the required <probability> <continuity equation>. The conserved Klein-Gordon density is proportional to
$$
\rho_{\rm KG}=i\left(\Psi^*\partial_t\Psi-(\partial_t\Psi^*)\Psi\right),
$$
up to a positive conventional constant. A pure positive-frequency mode gives $+2\omega|\Psi|^2$ and a negative-frequency mode gives $-2\omega|\Psi|^2$. \b[This density is not positive definite and cannot be a universal position <probability density>.]
For a free field, one can choose a positive-frequency subspace with a positive integrated inner product. Time-dependent backgrounds generally mix it with the negative-frequency subspace, so simply deleting the latter is not dynamically consistent. In <quantum field theory> the complex <Klein-Gordon field> instead has <particle> and <antiparticle> sectors; its <conserved current> is interpreted as charge current, and creation or annihilation of pairs need not fit a fixed-particle-number <wavefunction>. These are limitations of the naive single-particle interpretation, not a failure of the relativistic field equation.
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