Barnett talks about "neural engagement", but this is not a technical term. Googling around led to his patent on measuring so-called engagement. The relevant part is as follows:
“For example, if a movie was presented to a group of people, the measure of engagement could show the level of engagement the group (or a subset of the group) displayed in response to different scenes in the movie; the measure of engagement could also show how engaging the movie was overall. The method 100 preferably performs cross-brain correlations of neural data, calculated across pairs (a measure of neural similarity), as input for the measure of engagement. The method 100 additionally may function to provide a measure of engagement across small and precise time ranges. Understanding that one characteristic of engaging content is its ability to generate similar neural responses in different individuals, this preferably enables the method 100 to operate without the need to specify a model for the neural processes of engagement.”
So as far as I can tell, the fact that Trump generated higher levels of engagement means the EEG responses he elicited in viewers were more correlated with each other than were the EEG responses elicited by other candidates. This could potentially be interesting, but not without a process model explaining why. Even taking this associative, non-experimental method at face value, here's a plausible hypothesis that would render this result totally uninteresting: Everyone has seen and heard Donald Trump a lot. The same cannot be said for, say, John Kasich. It seems reasonable to me that frequent stimuli would be more likely to elicit common responses.
Maybe this hypothesis is correct; maybe it's not. The point is that without doing the hard work of showing they understand what their analytic technique measures, the results are totally uninterpretable. You can't even say that "Viewers weren't bored" without knowing what the correlations between the EEG responses of bored people would generate!
tl;dr: A poorly-designed and as-yet unpublished EEG study leads to an uninterpretable result that generated news coverage because readers like it when their latent beliefs are covered with a veneer of scientific acceptability.
(Professional quibble with the write-up: The term "lights up the brain" is neuroscientific slang used exclusively with methods like fMRI that tell you which regions of the brain are active. I know no neuroscientist who would say the brain is "lit up" based on an EEG reading.)