How to update a table by count from another table?
You can use JOIN syntax in UPDATE and conditional count in the subquery

UPDATE country_statistics s CROSS JOIN
(
  SELECT SUM(CASE WHEN list LIKE '%United States%' THEN 1 ELSE 0 END) us,
         SUM(CASE WHEN list LIKE '%United Kingdom%' THEN 1 ELSE 0 END) uk,
         SUM(CASE WHEN list LIKE '%Canada%' THEN 1 ELSE 0 END) ca
    FROM entries 
) q
   SET s.us = q.us, s.uk = q.uk, s.ca = q.ca
 WHERE id = 1


Here is SQLFiddle demo