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Eric:

The format of the tables produced by the USNO website is exactly the same as that used in the old Interactive Computer Ephemeris (ICE), a DOS program that predates MICA.  Odds are they used the same code.  The software predates the Nautical Almanac's switch from the Garfinkel model of refraction (horizontal refraction = 34.5') to Hohenkerk & Sinclair model (33.8').  Hohenkerk (I'm guessing) updated the coefficients in the old Bennet formula (to 7.32 and 4.32) to better match the new refraction model.

But those are formulae and models for removing the effect of refraction from apparent altitudes.  Instead, what's happening here is adding refraction to computed altitudes.  For that, the USNO modified the Bennet formula with a different set of coefficients, 8.6 and 4.4.  They first appear buried in sample FORTRAN code in USNO Circular No. 171.  (Sample BASIC code in the same paper uses 8.59 and 4.42 for reasons that were never explained; the FORTRAN coefficients seem to have stuck in future literature.)  The paper was published in 1987 (back when "FORTRAN" and "BASIC" were written in all-caps), back when Garfinkel was still the standard, so that is what they were probably trying to approximate.

My guess is that's how the predicted refraction is calculated.