Hi Blake,
The esriRequest makes it possible to request data, yes, but it is not clear where to request the data.
I have solved it by requesting geojson data directly from the Feature Service. Luckily, the layer is publicly available, so I don't need to worry about tokens. Using the terraformer/arcgis library to convert arcgis JSON to geojson is way too slow.
The geojson data CAN be converted with shp-write, however, you need to use the fixes done here, in order to export polygons: https://github.com/hwbllmnn/shp-write/tree/master/src
It would be nice, if an export feature was available in the JS API right out of the box, but it obviously isn't. People expect to be able to work with shapefiles in ESRI products. 🙂