To convert your object in JSON with Jackson:

import com.fasterxml.jackson.databind.ObjectMapper; 
import com.fasterxml.jackson.databind.ObjectWriter; 

ObjectWriter ow = new ObjectMapper().writer().withDefaultPrettyPrinter();
String json = ow.writeValueAsString(object);

Why not use GeoTools to handle the input and output? Then your code becomes simply something like:

File inFile = new File("/home/ian/Data/states/states.shp");
File outFile = new File("/home/ian/Data/states/states.geojson");
if (outFile.exists()) {
  outFile.delete();
}
outFile.createNewFile();
// Read
DataStore inputDataStore = DataStoreFinder.getDataStore(
        Collections.singletonMap("url", URLs.fileToUrl(inFile)));

String inputTypeName = inputDataStore.getTypeNames()[0];
SimpleFeatureType inputType = inputDataStore.getSchema(inputTypeName);

FeatureSource<SimpleFeatureType, SimpleFeature>
        source = inputDataStore.getFeatureSource(inputTypeName);

FeatureCollection<SimpleFeatureType, SimpleFeature>
        inputFeatureCollection = source.getFeatures();

Map<String, Object> params = new HashMap<>();
params.put(GeoJSONDataStoreFactory.URLP.key, URLs.fileToUrl(outFile));
// Write
DataStore newDataStore = DataStoreFinder.getDataStore(params);

newDataStore.createSchema(inputType);
String typeName = newDataStore.getTypeNames()[0];

SimpleFeatureStore featureStore =
        (SimpleFeatureStore) newDataStore.getFeatureSource(typeName);

featureStore.addFeatures(inputFeatureCollection);
newDataStore.dispose();
inputDataStore.dispose();

}

If you don't want to export all the attributes of the shapefile features you need to modify newDataStore.createSchema(inputType); to use a new limited schema. This needs something like:

SimpleFeatureType outType = DataUtilities.createSubType(inputType, new String[] {"type","REFCAT"});
newDataStore.createSchema(outType);