The minimum number of clues for a puzzle of $n^2$ x $n^2$ is known for $n = 2$ (4 x 4 Sudoku), of which the minimum clues is 4 (25% of cells filled). There are 13 such non-equivalent puzzles. (based on info at "enjoysudoku.com").
The minimum number of clues for a puzzle of $n^2$ x $n^2$ is also known for $n = 3$ (9 x 9 Sudoku), of which the minimum clues is 17 (about 21.0% of cells filled). About 49,000 such non-equivalent Sudokus have been discovered.
For $n = 4$ (16 x 16), the fewest clues known in any puzzle is 55 (about 21.5% of cells filled), but it is not known if this is the fewest possible. (based on info at "enjoysudoku.com").
For $n = 5$ (25 x 25), little work has also been done to find the fewest clues possible, and comparatively few Sudokus of this size have been constructed. One 25 x 25 Sudoku is known to have 328 clues. So from a cursory study, the fewest clues for (n=5) is 328 or fewer (about 52% or fewer cells filled). Note: this Sudoku was not an attempt to be an example with few clues. I listed it only to provide an initial upper bound.