Least-squares finite element methods for the inviscid Burgers equation in space-time domains are presented. Numerical results show that convergence of a standard least-squares approach for div(u,u2/2)=0 with bilinear elements for u and Newton linearization is problematic, possibly due to difficulties with the linearizability of the Burgers operator around discontinuous solutions. Alternative H(div)-conforming formulations are proposed in terms of the flux variables (or their De Rahm-dual) using face and edge elements. The face element formulation does not exhibit exact numerical conservation at the discrete level, but it is shown that the formulation converges to a conservative weak solution with the correct shock speed. The dual edge element formulation is strictly conservative at the discrete level. The least-squares functional naturally provides a sharp a posteriori error estimator that is used for adaptive refinement in space-time. Standard algebraic multigrid methods are applied to the positive semi-definite matrices resulting from the new least-squares formulations, and multigrid efficiency as a function of problem size is investigated. Extension of the new least-squares methods to general systems of hyperbolic conservation laws in multiple dimensions is discussed.