Gibbons-Hawking boundary for electrodynamics?

Can you compute a Lagrange-density for Maxwell-electrodynamics that depends on a second instead of squares of first derivatives of the potential \(A_\mu\), through integration by parts? Would there be an analogous Gibbons-Hawking term as in gravity, when computing the Einstein-Hilbert action from the Einstein-Palatini action? Would a gauge condition such as \(\eta^{\mu\nu}\partial_\mu A_\nu = 0\) have implications for the action and the possible boundary term?