group by month of unix timestamp field

SELECT 
  MONTH(FROM_UNIXTIME(date_assigned)), 
  YEAR(FROM_UNIXTIME(date_assigned)), 
  COUNT(*)
FROM assignments
GROUP BY 
  MONTH(FROM_UNIXTIME(date_assigned)), 
  YEAR(FROM_UNIXTIME(date_assigned))

Your date_assigned column should be of type DATE. AFAIK MONTH works on date columns and if you want the month name from a DATE column use : MONTHNAME(date_assigned)