Solution found with 19 lines.
Line 1 starts at (0,0)
  goes through (0,0) (0,1) 
  and ends at (0,1)
Line 2 starts at (0,1)
  goes through (0,1) (1,0) 
  and ends at (1,0)
Line 3 starts at (1,0)
  goes through (0,0) (1,0) 
  and ends at (0,0)
Line 4 starts at (0,0)
  goes through (0,0) (1,1) 
  and ends at (1,1)
Line 5 starts at (1,1)
  goes through (0,1) (1,1) 
  and ends at (0,1)
Line 6 starts at (0,1)
  goes through (0,1) (0,2) 
  and ends at (0,2)
Line 7 starts at (0,2)
  goes through (0,2) (1,0) 
  and ends at (1,0)
Line 8 starts at (1,0)
  goes through (0,3) (1,0) 
  and ends at (0,3)
Line 9 starts at (0,3)
  goes through (0,0) (0,1) (0,2) (0,3) 
  and ends at (0,0)
Line 10 starts at (0,0)
  goes through (0,0) (1,2) 
  and ends at (1,2)
Line 11 starts at (1,2)
  goes through (0,1) (1,2) 
  and ends at (-2,-1)
Line 12 starts at (-2,-1)
  goes through (0,0) (2,1) 
  and ends at (2,1)
Line 13 starts at (2,1)
  goes through (0,2) (2,1) 
  and ends at (0,2)
Line 14 starts at (0,2)
  goes through (0,2) (2,3) 
  and ends at (3,7/2)
Line 15 starts at (3,7/2)
  goes through (3,0) (3,1) (3,2) (3,3) 
  and ends at (3,-1/2)
Line 16 starts at (3,-1/2)
  goes through (0,1) (2,0) 
  and ends at (-6,4)
Line 17 starts at (-6,4)
  goes through (0,4) (1,4) (2,4) (3,4) (4,4) 
  and ends at (4,4)
Line 18 starts at (4,4)
  goes through (4,0) (4,1) (4,2) (4,3) (4,4) 
  and ends at (4,0)
Line 19 starts at (4,0)
  goes through (0,4) (1,3) (2,2) (3,1) (4,0) 
  and ends at (0,4)

Solution found with 19 lines.
This solution is reentrant.
Line 1 starts at (0,0)
  goes through (0,0) (0,1) 
  and ends at (0,1)
Line 2 starts at (0,1)
  goes through (0,1) (1,0) 
  and ends at (1,0)
Line 3 starts at (1,0)
  goes through (0,0) (1,0) 
  and ends at (0,0)
Line 4 starts at (0,0)
  goes through (0,0) (1,1) 
  and ends at (1,1)
Line 5 starts at (1,1)
  goes through (0,1) (1,1) 
  and ends at (0,1)
Line 6 starts at (0,1)
  goes through (0,1) (0,2) 
  and ends at (0,2)
Line 7 starts at (0,2)
  goes through (0,2) (1,0) 
  and ends at (1,0)
Line 8 starts at (1,0)
  goes through (0,3) (1,0) 
  and ends at (0,3)
Line 9 starts at (0,3)
  goes through (0,0) (0,1) (0,2) (0,3) 
  and ends at (0,0)
Line 10 starts at (0,0)
  goes through (0,0) (1,2) 
  and ends at (1,2)
Line 11 starts at (1,2)
  goes through (0,1) (1,2) 
  and ends at (-2,-1)
Line 12 starts at (-2,-1)
  goes through (0,0) (2,1) 
  and ends at (2,1)
Line 13 starts at (2,1)
  goes through (0,2) (2,1) 
  and ends at (0,2)
Line 14 starts at (0,2)
  goes through (0,2) (2,3) 
  and ends at (3,7/2)
Line 15 starts at (3,7/2)
  goes through (3,1) (3,2) (3,3) 
  and ends at (3,1)
Line 16 starts at (3,1)
  goes through (0,4) (1,3) (2,2) (3,1) 
  and ends at (0,4)
Line 17 starts at (0,4)
  goes through (0,4) (1,4) (2,4) (3,4) (4,4) 
  and ends at (4,4)
Line 18 starts at (4,4)
  goes through (4,0) (4,1) (4,2) (4,3) (4,4) 
  and ends at (4,0)
Line 19 starts at (4,0)
  goes through (0,0) (1,0) (2,0) (3,0) (4,0) 
  and ends at (0,0)

Solution found with 18 lines.
Line 1 starts at (0,0)
  goes through (0,0) (0,1) 
  and ends at (0,1)
Line 2 starts at (0,1)
  goes through (0,1) (1,0) 
  and ends at (1,0)
Line 3 starts at (1,0)
  goes through (0,0) (1,0) 
  and ends at (0,0)
Line 4 starts at (0,0)
  goes through (0,0) (1,1) 
  and ends at (1,1)
Line 5 starts at (1,1)
  goes through (0,1) (1,1) 
  and ends at (0,1)
Line 6 starts at (0,1)
  goes through (0,1) (0,2) 
  and ends at (0,2)
Line 7 starts at (0,2)
  goes through (0,2) (1,0) 
  and ends at (1,0)
Line 8 starts at (1,0)
  goes through (0,3) (1,0) 
  and ends at (0,3)
Line 9 starts at (0,3)
  goes through (0,0) (0,1) (0,2) (0,3) 
  and ends at (0,0)
Line 10 starts at (0,0)
  goes through (0,0) (1,2) 
  and ends at (1,2)
Line 11 starts at (1,2)
  goes through (0,1) (1,2) 
  and ends at (-2,-1)
Line 12 starts at (-2,-1)
  goes through (0,0) (2,1) 
  and ends at (2,1)
Line 13 starts at (2,1)
  goes through (2,1) (3,2) (4,3) 
  and ends at (5,4)
Line 14 starts at (5,4)
  goes through (3,0) (4,2) 
  and ends at (2,-2)
Line 15 starts at (2,-2)
  goes through (2,0) (2,