Appending (pushing) and removing from a JSON array in PostgreSQL 9.5+

To add the value use the JSON array append opperator (||)

UPDATE jsontesting
SET jsondata = jsondata || '["newString"]'::jsonb
WHERE id = 7;

Removing the value looks like this

UPDATE jsontesting
SET jsondata = jsondata - 'newString'
WHERE id = 7; 

Concatenating to a nested field looks like this

UPDATE jsontesting
SET jsondata = jsonb_set(
  jsondata::jsonb,
  array['nestedfield'],
  (jsondata->'nestedfield')::jsonb || '["newString"]'::jsonb) 
WHERE id = 7;