A bug travels in the coordinate plane moving only along the lines...

Least number of steps required to move from $A$ to $B$ is 10 . Let we denote any step towards positive $x$ axis by $R$ and negative $y$-axis by $D$.
So all steps are the permutations of $R R R R R R D D D$. In this way, all points of the rectangular grid are covered. If we introduce one step in negative $x$ axis or positive $y$ axis, movement will be completed in 12 steps. Similarly, if we introduce two steps in negative $x$-axis or in positive $y$-axis or one in each direction the movement will be completed in 14 steps (maximum allowed).
Maximum number of points with integer coordinates $=9 \times 7+7 \times 2+5 \times 2=87$