Dissection puzzle for area 49 to area 50 49 and 50 are close, as are 288 and 289.  That allows a grid illusion. If cut out of wood, perhaps with coloring on the border as an "assistance", the pieces could be dumped out of the tray, flipping them, then scrambled a bit, and the solver could be asked to put them back into the tray.  But there would be a gap.  "No, that's not right."  The owner would dump the pieces out again, then place them in perfectly, then dump them again, and invite the solver to try again.
Are there better dissections that can be used, which stay on the grid lines?  Ideally, I'd like to stay under 7 pieces, of roughly the same area, and I'd like the hole in the second figure to be square. 

Joseph Kisenwether sent me the following, which is pretty good.

I'm not looking for old versions of the Missing Square Puzzle.  I'm looking for new dissections specifically for squares with area 49 and 50, or areas 288 and 289, and where all dissection lines are in the directions of queen moves.
 A: 
ver.1 : 7 pieces / 6, 6, 6, 6.5, 8, 8, 8.5
ver.2 : 6 pieces / 7, 7.5, 8, 8, 8.5, 10
ver.3 : 6 pieces / 7, 8, 8, 8.5, 8.5, 9
These are my designs for the dissection puzzle 49 to 50. They have same idea that the boundary pieces in the square of area 50 are put together to make the diagonal line in the square of area 49.
Ver.1 is the first design I made, but as I confused that 'stay under 7' means 'less than 7' or not, I made second version of design. Ver.3 is better than Ver.2 in the sence of 'roughly same area', but in actual puzzle, Ver.2 is better than Ver.3 because people can solve Ver.3 much faster than Ver.2.(I'm sure that people will put together yellow and blue piece first in solving Ver.3, which leads to the goal.)
A: I guess the following might be also in the Wikipedia article Missing square puzzle, but A. Beutelspacher explained this very well in his book "Diskrete Mathematik für Einsteiger" (ISBN 978-3-8348-1248-3)
I am not sure if this is what you're looking for, but it might still be interesting.
Fibonacci numbers
Fibonacci numbers are integers that are defined as
$$f_0 := 0, f_1 := 1, \forall n \in \mathbb{N}_{\geq 2}:f_n := f_{n-1} + f_{n-2}$$
The first fibonacci-numbers are
$$0,1,1,2,3,5,8,13,21, \dots$$
Simpson-Identity
The Simpson-Identity states that
$$\forall n \in \mathbb{N}_{\geq 2}: f_{n+1} \cdot f_{n-1} - f_n^2 = (-1)^n $$
Proof: with induction
Missing square puzzle

(I've added this as SVG to Wikipedia, just in case you want to see this bigger: Missing-squre-fibonacci.svg)
So, whats going on here?
The square on the left has a size of $f_n^2$, the rectangle on the right seems to have a size of $f_{n-1} \cdot f_{n+1}$ which makes (according to Simpson's identity) a difference of $\pm 1$. Now you can make $n$ as big as you want, the difference will still be $\pm 1$. This means, you can make the difference as difficult to see as you want.
A: Have you ever seen the Missing Square Puzzle?

Maybe this isn't exactly what you're looking for, but it is along the same lines.  At the bottom of the wikipedia article, you'll find similar puzzles (like Sam Lloyd's paradoxical disection), which also may help.  Here is one that is fairly interesting.

