There are large populations of Amish who don't use cell phones, landline phones, or anything. The closest they'd get to a phone call is asking a neighbour to call 911 in an emergency (assuming they're even willing to do that).

One group I am aware of will only use a payphone in the nearest town. They actually filed to force AT&T to keep a payphone there because the relevant tariff required AT&T to do so, and were the only people who ever bothered to make AT&T do this. So there is one payphone in that town that they go to and drop their quarters in to make phone calls.

There are no "secret" cell phones there.