What I am needing is:
Return the sum of the count from Column G that contains a date of
1/1/2050, and a count for Column H that is equal to today().

Not sure exactly what's required.

If you need two sums, then
(1) the count from Column G that contains a date of 1/1/2050 is
=COUNTIF($G:$G,DATE(2050,1,1))
and
(2) the count for Column H that is equal to today() is
=COUNTIF($H:$H,TODAY())

If you need the total of these two, add the two formulas.

Notice that if you add the two formulas then any row that satisfies both (1) and (2) will be counted twice. If you want to count such rows only once, then this should work:
=COUNTIF($G:$G,DATE(2050,1,1))+
COUNTIF($H:$H,TODAY())-
SUMPRODUCT(--($G:$G=DATE(2050,1,1)),--($H:$H=TODAY()))

Hope this helps getting started.