The issue can be tackled from a completely mathematical perspective. What you are saying about $\phi^3$ is true at classical level. The classical Hamiltonian functional ${\cal H}(\phi,\partial\phi)$ is not bouded below because we can find field configurations $(\phi_0, \partial \phi_0)$ with $-{\cal H}(\phi_0, \partial \phi_0)$ arbitrarily large.
and this gives rise to several pathologies of various nature.

The issue with the hydrogen atom is mathematically different. There you may looks for wavefunctions $\psi$ such that $\langle \psi|H\psi\rangle$ is arbitrarily cose to $-\infty$. 

However, differently from the (classical) Klein-Gordon $\phi^3$, here you also have the constraint $$\langle \psi|\psi\rangle = \int_{\mathbb{R}^3} |\psi(\vec{x})|^2 d^3x <+\infty\:.$$
This constraint prevents for finding functions with $$-\langle \psi|H\psi\rangle$$ arnbitrarily large. The proof is the (well known) computation of the spectrum of the hydrogen atom.
 
One can also consider the second quantization of $\phi$. In that case, after normal ordering renormalization one sees that it is however possible to find states $\Psi$  of the quantum field where $-\langle \Psi| \hat{\cal H} \Psi\rangle$ is arbitrarily large, in spite of the constraint $\langle \Psi| \Psi\rangle=1$, due to the nature of $\hat{\cal H}$ written in terms of cration and annihilation operators.