Here a path refers to a series of connected segments. I have tried out small cases and have conjectured that the minimum is $4N$ and the maximum is $2N^2+2N+1$.
Construction for minimum:
Construction for maximum:
However, I do not know how to prove that these are the minimum/maximum. Any help will be appreciated!
Minimum:
The $8N$ edges of the board give $8N$ path endpoints, so there are at least $4N$ paths. The first example has $4N$ paths, so that is indeed the minimum.
Maximum:
The total length of the paths is $2(2N)^2$ segments, since each square has two segments. To get the maximum number of paths out of these segments, the paths need to be as short as possible.
The end points at the edge of the board can have as their shortest paths four paths of length 1 at the corners of the board, but all the other edges will need paths of length at least two because they have to start and end in different squares. The shortest possible loop is of length 4, a small diamond square.
The best possible case is therefore when $8$ of the $8N$ end points are used up by paths of length 1, and the remaining $8N-8$ endpoints as paths of length 2. That would give $4+(4N-4)$ paths with a total length of $8N-4$. The remaining $8N^2-8N+4$ segments would at best form $2N^2-2N+1$ loops of length 4. Together that would give at most $4+(4N-4)+(2N^2-2N+1) = 2N^2+2N+1$ paths.
The second example shows that this is indeed possible.