All I can think of using a bunch of IF()'s and adding them together:
SELECT IF(col1 IS NOT NULL, 1, 0) + IF(col2 IS NOT NULL, 1, 0) + IF(col3 IS NOT NULL, 1, 0) AS non_empty
FROM table . . .
If a column should be counted as empty if it's an empty string (instead of actually NULL), then you would have to make it even uglier with:
IF(col1 IS NOT NULL AND col1 != '', 1, 0)