We do not need to generalize the EPR paradox to relativistic quantum mechanics/field theory, as the only assumption that is needed from relativity is that information travels at a finite speed, due to which we can in principle construct spacelike intervals, which is necessary for formulating the paradox. You can in principle take two non-commuting field operators and do the same thing, but that will not give you anything extra apart from the non-relativistic formulation.
The paradox as formulated in the original paper, was intended to show that quantum mechanics was not a physically complete theory. The intended motivation was that you can measure two different non-commuting observables simultaneously on spacelike separated particles, and since the information can't travel faster than the speed of light, there must be some kind of classical local hidden variables underlying quantum mechanical description which can explain these correlations.
The paradox led to formulation of Bell's inequalities for verifying whether the correlations in a system are classical or quantum. Experimentally classical correlations have been ruled out, and that means nature can't be described by a local hidden variable theory.