Other than the excess heat, tritium, X-rays, and gamma rays, of course.
None are credible or (importantly) commensurate.
"Wouldn’t a nuclear reactor help out by producing it’s own additional neutrons, chain reaction style?"
If they were relying on chain reactions in U-235 e.g., they wouldn't need LENR.
The idea is that fast neutrons can cause fission of the much more abundant U-238 (or even Th-232). However, unlike U-235, U-238 and Th-232 do not sustain chain reactions.