Is there a relationship between Free Radicals and Beta Particles?
There is high-energy in the sense of chemical reactivity, and then there is high energy in the sense of nuclear reactivity. The energy associated with the latter is many orders of magnitude greater than the energy associated with the former, so that there is generally no direct interaction between the two (where the free radical is chemically reactive, and the beta particle arises from nuclear reactivity).  

A free radical is a chemical species, whose electron configuration is such that it is highly reactive towards other chemical species. However, this energy is far too low to affect any nuclear reactivity.  

A beta particle is a high-energy electron or positron emitted by the radioactive decay of an atomic nucleus. As these particles are emitted from the nucleus of the atom, the electron configuration (i.e. a free radical) is irrelevant; only the stability/reactivity of the nucleus determines whether an atom will emit a beta particle, not the chemical species.  

If any of this seems incorrect or unclear, please do not hesitate to ask for clarifications in the comments.