As Jetse Stoorvogel points, the "Band Collection Statistics" in the "Spatial Analyst Tools" computes summary statistics, matrix of covariance and correlation, the documantatios is available in:
The former are not autocorrelation indices, it's just the bivariate Pearson index, if you need to compute spatial autocorrelation indices (global Moran's I or th Getis Ord Gi*) you will need to transform your raster data to points (to geodatabase, because it will be massive!) and use the modelling spatial relationships and analyzing patterns tools within the "Spatial Statistics Tools"
As Dr. Caudillo said, if the images are not too big, you can transform the raster data to points as txt file, and them using a statistical software you can obtain what do you need.
I think you want the Band Collection Statistics tool in Spatial Analyst. It takes rasters as input and can output the correlation matrix (among other statistics). Just make sure to check the "Compute covariance and correlation matrices" box.
"even with the paths scratched out, they appear very long, on some sort of network drive... can you not work on your local machine with a simple folder structure (c:\test for example) to try to rule out their location and path issues "
You could also make a systematic point dataset, and extract the values from each raster into separate columns. Then you could calculate the correlation coefficient between the separate rasters in Excel.