Par4QC I don't think that there are any specific requirements to be captain (on either side), but I suspect that it's expected that 1) the captain should meet the same eligibility criteria as the players, 2) the captain has played in several Ryder Cups with a reasonable degree of success, 3) the captain is reasonably popular on Tour, and 4) the captain is likely to be competent at all aspects of the job.
One thing that doesn't seem to be stated explicitly (but is implicit, on both sides) is that any player/captain needs to be from either the US or from Europe. For example, Brooks Koepka was never eligible for the European Ryder Cup team in 2014 despite playing primarily in Europe that year and finishing in the top 10 of that year's Race to Dubai.
