So coming from someone that deals with electricity, generally electricity is considered the phenomena associated the the presence and motion of electric charges. Anyways for electricity to flow and be produced there needs to be a non-0 net charge in motion, however (I may be incorrect here) when a nerve fires it moves relatively charged ions in and out of the neurons but ultimately has a net 0 motion of charge... As in there are an equal number of positive and negative charges in motion, and things are only truly charged relative to each other. The transfer of energy (at least from a high school biology class) is caused by using energy (ATP) to actively transport ions against the diffusion gradient and when the nerve "fires" it allows everything to passively diffuse to equilibrium.
Thus there is a relative voltage and charge differential but it is not electricity in some sense.
Now I can be completely wrong here (only taken high school biology) but this would be how I would attempt to formulate an argument to support a stance of "the human body doesn't produce electricity."
However, that being said there are actual non-0 charge transfers in the human body (aerobic respiration for example, aka krebs cycle) and thus uses a chemical potential to generate an electric current. But shhhh