  We present a novel methodology for deriving high-order quadrature rules
(HOSQ) designed for the integration of scalar functions over regular embedded
manifolds. To construct the rules, we introduce square-squeezing--a
homeomorphic multilinear hypercube-simplex transformation--reparametrizing an
initial flat triangulation of the manifold to a hypercube mesh. By employing
square-squeezing, we approximate the integrand and the volume element for each
hypercube domain of the reparameterized mesh through interpolation in
Chebyshev-Lobatto grids. This strategy circumvents the Runge phenomenon,
replacing the initial integral with a closed-form expression that can be
precisely computed by high-order quadratures. We prove novel bounds of the
integration error in terms of the $r^\text{th}$-order total variation of the
integrand and the surface parameterization, predicting high algebraic
approximation rates that scale solely with the interpolation degree and not, as
is common, with the average simplex size. For smooth integrals whose total
variation is constantly bounded with increasing $r$, the estimates prove the
integration error to decrease even exponentially, while mesh refinements are
limited to achieve algebraic rates. The resulting approximation power is
demonstrated in several numerical experiments, particularly showcasing
$p$-refinements to overcome the limitations of $h$-refinements for highly
varying smooth integrals.
