1. Key idea
   For a particle moving in the $x$-$y$ plane, angular momentum about the origin is

Since motion is in the plane, $\vec L$ will be along $\hat k$.

Also, for motion along a straight side with constant velocity, the magnitude of angular momentum about the origin is

with sign determined by $\vec r\times \vec v$.

2. Geometry of the square
   From the given expressions, the square is centered such that each side is at perpendicular distance involving $\dfrac{R}{\sqrt 2}$ and shifted by $a$.
   The distance of the line through side $AB$ and side $CD$ from the origin differs by $a$, and similarly for the vertical sides.

Let us determine the angular momentum side by side using sign conventions.

3. When particle moves from $A$ to $B$
   This is along one horizontal side. The perpendicular distance of this line from origin is

The sense of motion gives clockwise rotation about origin, hence

So option C is correct.

4. When particle moves from $D$ to $A$
   This is along a vertical side. Its perpendicular distance from origin is also

Now the direction gives positive $\hat k$, so

So option B is correct.

5. When particle moves from $B$ to $C$
   This is the opposite horizontal side, displaced by side length $a$. Hence perpendicular distance becomes

not $\dfrac{R}{\sqrt 2}+a$.

Its sign is positive, therefore

So option A is false.

6. When particle moves from $C$ to $D$
   This is the opposite vertical side, displaced by side length $a$. The perpendicular distance should become

or with sign from direction, equivalently the given expression must be checked carefully.

For this side, the direction gives negative contribution, so

Thus the expression

is not correct.

So option D is false.

7. Conclusion
   The false statements are:

8. Comparison with stored answer
   Stored correct answer: $B, D$

My derived answer: $A, D$

They do not match. Option $B$ is actually correct because for motion along $D \to A$, the perpendicular distance from origin is $\dfrac{R}{\sqrt 2}$ and the sign is positive, giving

Hence the stored answer appears to incorrectly include $B$ and miss $A$.