  We present the robust selection of quiescent (QG) and post-starburst (PSB)
galaxies using ultra-deep NIRCam and MIRI imaging from the JWST Advanced Deep
Extragalactic Survey (JADES). Key to this is MIRI 7.7$\mu$m imaging which
breaks the degeneracy between old stellar populations and dust attenuation at
$3<z<6$ by providing rest-frame $J$-band. Using this, we identify 23 passively
evolving galaxies in UVJ color space in a mass-limited (log
$M_{\star}/M_{\odot}\geq8.5$) sample over 8.8 arcmin$^2$. Evaluation of this
selection with and without 7.7$\,\mu$m shows that dense wavelength coverage
with NIRCam ($8-11$ bands including $1-4$ medium-bands) can compensate for
lacking the $J-$band anchor, meaning that robust selection of high-redshift QGs
is possible with NIRCam alone. Our sample is characterized by rapid quenching
timescales ($\sim100-600$ Myr) with formation redshifts $z_{\rm f}\lesssim8.5$
and includes a potential record-holding massive QG at $z_{\rm
phot}=5.33_{-0.17}^{+0.16}$ and two QGs with evidence for significant residual
dust content ($A_{\rm V}\sim1-2$). In addition, we present a large sample of 12
log $M_{\star}/M_{\odot}=8.5-9.5$ PSBs, demonstrating that UVJ selection can be
extended to low mass. Analysis of the environment of our sample reveals that
the group known as the Cosmic Rose contains a massive QG and a dust-obscured
star-forming galaxy (a so-called Jekyll and Hyde pair) plus three additional
QGs within $\sim20$ kpc. Moreover, the Cosmic Rose is part of a larger
overdensity at $z\sim3.7$ which contains 7/12 of our low-mass PSBs. Another 4
low-mass PSBs are members of an overdensity at $z\sim3.4$; this result strongly
indicates low-mass PSBs are preferentially associated with overdense
environments at $z>3$.
