Does E=mc^2 Require Relativity?

Physicist Tony Rothman has a new paper arguing that one can get the famous Einstein mass-energy equivalience E = mc² before Einstein's 1905 paper, and without relativity. In particular, it appears in a 1900 Poincare paper.

Many physicists, for instance, are under the impression that ℰ=m⁢c² can be established by employing the four-vector formalism of special relativity. An early draft of Wikipedia’s page on mass-energy equivalence in fact offered exactly such a “derivation.” Four-vectors, however, are defined in order to be consistent with ℰ=m⁢c²; consequently any argument based on them to prove the relationship is circular. ...

A universal, assumption-free proof of ℰ=m⁢c² is no more attainable than a universal proof of conservation of energy or momentum, and the very idea that all physics can be derived from a master Lagrangian without experimental input must be doomed to failure. For that reason, all demonstrations of mass-energy equivalence rely on specific assumptions and approximations. The closest thing that exists to a general proof of ℰ=m⁢c² is the Laue-Klein theorem [16, 17, 18] of 1911 and 1918, which in essence states that if ℰ=m⁢c² holds for a point mass, then it also holds for an extended closed system, under specified boundary conditions. If radiation can escape to infinity, for example, the boundary conditions are evaded.

Einstein was aware of the inadequacies of his 1905 article and attempted to correct them in six further papers, but as Ohanian argues [19], none is free of errors and inconsistencies. Physicists who have actually read the 1905 paper know that the dubious step is the final one, in which Einstein relies on the Newtonian value for the kinetic energy. ...

Can one arrive at ℰ=m⁢c² in a consistent and plausible manner using only Galilean mechanics and “perhaps Maxwellian” electrodynamics?

Okay, but Maxwellian electrodynamics is a fully relativistic theory, if interpreted correctly. The whole theory of special relativity is mostly a recognition of that fact.