Does the brain stop processing data when the eyes move? I was reading this question and remember back to the 70s when our local newspaper changed formats. In an article about why they made the changes that they did, one of the points they mentioned was that by making the columns narrower, you would retain more because the brain didn't process the words when the eyes were moving. 

Over the past few years working in instructional design, I've often wanted to cite a reference for my newspaper's statement, or at the very least a reference that debunks it. I've been unsuccessful finding anything on google, but I'm not sure if I'm asking the right questions or not. It may very well be that this was a theory in the 70's, since disproven, junk science, or something very real for which I've not stumbled upon the right search terms. 

In any case, it seems to me that I'm the only person in the world whose ever heard this idea, and I'd like to find some reference, positive or negative, to it. Can anyone direct me to some research on the topic?
I'm sorry you happened upon that research from the 70s. Should it have been true those with Nystagmus (constant involuntary eye movement) would never learn anything so it must be in a general sense false.
Here is a current thoughts and research into the subject:

Eye movement-related brain activity during perceptual and cognitive processing
For several decades researchers have been recording electrical brain activity associated with eye movements in attempt to understand their neural mechanisms. However, recent advances in eye-tracking technology have allowed researchers to use eye movements as the means of segmenting the ongoing brain activity into episodes relevant to cognitive processes in scene perception, reading, and visual search. This opened doors to uncovering the active and dynamic neural mechanisms underlying perception, attention and memory in naturalistic conditions. The present eBook contains a representative collection of studies from various fields of visual neuroscience that use this cutting edge approach of combining eye movements and neural activity.
The majority of the articles in the eBook combine the measurement of eye movements with the recording of the electroencephalogram (EEG) in human subjects performing various psychological tasks. The most common methodological approach is examination of the EEG activity time-aligned to certain eye movement events, such as the onset of a fixation or a start of a saccadic eye movement (Fischer et al., 2013; Frey et al., 2013; Henderson et al., 2013; Hutzler et al., 2013; Nikolaev et al., 2013; Richards, 2013; Simola et al., 2013). Several works employ the time-frequency and synchrony analysis (Fischer et al., 2013; Hoffman et al., 2013; Ito et al., 2013; Nakatani and Van Leeuwen, 2013; Nakatani et al., 2013).