The main reason for their exemptions is their demonstrated track record of self-sufficiency. They have a very strong sense of taking care of each other - there are no poor people in Amish communities. So the courts allowed them to opt out of Social Security, because they have shown they provide their own. Similarly with education - their system works with only a minimal formal education, so the courts were happy to leave them be at that point.

Basically they just fought both in court. I think their public perception is positive enough that the courts accepted it.

They are required to register for selective service. If drafted, they declare themselves to be a conscientious objector. This may include working in a hospital or other non-military role, but usually they request and get a farm deferment, to work on Amish or Mennonites farms, producing food as their "war effort."

See http://www.daviddfriedman.com/Academic/Course_Pages/Legal_Sy... for more details.

Basically, the Amish found that of the men who do non-combantant military service or work as nurse aides in hospitals end up being away from the Amish community for a couple of years, "only about half of them chose to return to their communities when their service was done, and not all of those chose to join the church." The Amish set up a committee to negotiate with the US about a solution which would work with both the US government and their own community.