Does FMLA require an employer to pay holiday pay to an employee while on FMLA if he/she would otherwise qualify for holiday pay?

FMLA is, by definition, unpaid job protection, so the whole premise of holiday pay does not apply.

At our company, we allow holiday pay to be taken while on paid leave (sick/annual, STD, or LTD), but do not allow it while on unpaid leave. Both are independent of FMLA status, since it doesn't apply.

Hope this helps!