  In this work we study the Higgs boson decays into two photons and into one
photon and one $Z$ gauge boson within the context of the non-linear Effective
Field Theory called the Electroweak Chiral Lagrangian. We present a detailed
computation of the corresponding amplitudes to one-loop level in the covariant
$R_\xi$ gauges. We assume that the fermionic loop contributions are as in the
Standard Model and focus here just in the computation of the bosonic loop
contributions. Our renormalization program and the anatomy of the various
contributions participating in the $R_\xi$ gauges are fully explored. With this
present computation we demonstrate the gauge invariance of the EChL result, not
only for the case of on-shell Higgs boson, but also for the most general and
interesting case of off-shell Higgs boson. We finally analyse and conclude on
the special relevance of the Goldstone boson loops, in good agreement with the
expected chiral loops behaviour in Chiral Lagrangians. We perform a systematic
comparison with the corresponding computation of the Standard Model in the
$R_\xi$ gauges and with the previous EChL results in the unitary gauge. This
work represents the first computation within the EChL of these Higgs
observables to one-loop in the most general $R_\xi$ gauges and with a full
renormalization program description, not yet fully explored in the previous
literature and which is different to the most frequently used in the linear
Effective Field Theory (SMEFT).
