Square Number Digit Density
Answering the bonus: the best score for numbers &lt;1e9 is 5/3=1.666..., generated by 144411449 (and maybe others?).

But you can do better with larger numbers.  Generally if n has a score of x, then you can concatenate two copies of n and get the same score x.  If you're lucky and n has the same first and last digit, then you can drop one of those digits in the concatenation and improve your score slightly (one less than double the number of squares and one less than double the number of digits).

n=11449441 has a score of 1.625 and has the same first &amp; last digit.  Using that fact, we get the following sequence of scores:

1.625 for 11449441
1.666 for 114494411449441
1.682 for 1144944114494411449441
1.690 for 11449441144944114494411449441
1.694 for 114494411449441144944114494411449441


which gives an infinite sequence of numbers which are strictly (although decreasingly) better than previous numbers, and all but the first 2 better than the best score for numbers &lt; 1e9.

This sequence may not be the best overall, though.  It converges to a finite score (12/7=1.714) and there may be other numbers with better scores than the limit.

Edit: a better sequence, converges to 1.75

1.600 14441
1.667 144414441
1.692 1444144414441
1.706 14441444144414441
1.714 144414441444144414441