  We study transverse equivariant Hilbert schemes of affine hypertoric
varieties equipped with a symplectic action of a Weyl group. In particular, we
show that the Coulomb branches of Braverman, Finkelberg, and Nakajima can be
obtained either as such Hilbert schemes or Hamiltonian reductions thereof.
Furthermore, we propose that the Coulomb branches for representations of
non-cotangent type are also obtained in this way. We also investigate the
putative complete hyperk\"ahler metrics on these objects. We describe their
twistor spaces and, in the case when the symplectic quotient construction of
the hypertoric variety is $W$-equivariant (which includes Coulomb branches of
cotangent type), we show that the hyperk\"ahler metric can be described as the
natural $L^2$-metric on a moduli space of solutions to modified Nahm's
equations on an interval with poles at both ends and a discontinuity in the
middle, with the latter described by a new object: a hyperspherical variety
canonically associated to a hypertoric variety.
