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When consulting Schwartz's QFT and Standard Model, I observe that he writes the kinetic term for field species in a form somewhat a little bit different from the ones appearing in other literature. For instance, we always write the kinetic term for a scalar field $\phi$ as $\partial^\mu\phi\partial_\mu\phi$, but sometimes Schwartz writes it as $\phi\Box\phi$. I understand they are the same up to a divergence term which is immaterial and we are merely doing integration by part. In the more common form, the Lagrangian is considered as a function of the field and its first derivative. In Schwartz's box form, the kinetic term contains second derivatives of the field. My question is: Should we consider the Lagrangian as a function of the field and its second derivative when using the box form of the kinetic term and use the following form of the Euler Lagrange equation to derive the classical equation of motion?

\begin{equation} \frac{\partial L}{\partial\phi}-\partial_{\mu}\frac{\partial L}{\partial\phi,_\mu}+\partial_\mu\partial_\nu\frac{\partial L}{\partial \phi_{,\mu\nu}}=0 \tag{1} \end{equation}

where $\phi,_\mu$ denotes the first derivatives of the field while $\phi_{,\mu\nu}$ denotes the second derivative of the field and this is the Euler Lagrange equation for a Lagrangian containing higher derivative which is derived from usual procedure of $\delta S=0$. (I tried to use it and apparently works, but not sure weather this is the correct way to understand the situation)

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Phycisists often write the Lagrangian density $${\cal L} ~\sim~-\phi\Box\phi$$ instead of $${\cal L} ~\sim~\partial^\mu\phi\partial_\mu\phi$$ merely because it is simpler to write. [Here the symbol '$\sim$' means equality modulo total spacetime divergence terms.]

If one insists on the former version with second derivatives, then OP is right that the Euler-Lagrange (EL) equations take the form of eq. (1).

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