Does a maximal inscribed square in a regular polygon have a side parallel to a side of the polygon? Suppose $P$ is a regular (i.e.,equiangular equilateral) polygon in the Euclidean plane, and the number sides of $P$ is not a multiple of $4$. Then $P$ contains an inscribed square. (Citation.) Of all the squares inscribed in $P$, some are largest. (Proof, the boundary of $P$ is compact, so a subsequence of vertices of increasing inscribed squares has a limit.) Among all such largest squares, does at least one of them have a side parallel to a side of $P$? It looks like it should, so I suspect it does, but that's not worth much.
 A: No. Consider regular octagon. Then largest inscribed square will be this one:
 
It's not hard to prove this. Consider the circumcircle of octagon. Then, largest square inside octagon cannot be larger than largest square inscribed in circle. But in that case, they're the same, so this square is the largest inside octagon.
A: Please consult this reference:
Dilworth, S. J.; Mane, S. R., On a problem of Croft on optimally nested regular polygons, J. Geom. 99, No. 1–2, 43–66 (2010). ZBL1231.68264.
Let P be a regular n-gon.
If n is even not divisible by 4, the paper shows that the largest inscribed square has edges parallel with edges of P.
(In this case the largest inscribed square is also the largest square contained in P.)
So suppose n is odd.
The case n=3 is well known (e.g. Martin Garder in Scientific American).
The inner square shares an edge with the outer triangle.
The above paper shows that an inscribed square shares a common symmetry axis with P.
Every symmetry axis of P passes through a vertex and bisects an opposite edge of P (because n is odd).
If that axis is a diagonal of the square, the square cannot have 4 points of contact with P
(it would have to share a vertex with P).
So the common symmetry axis must bisect opposite edges of the square.
Then the square has two edges parallel to an edge of P.
So the answer to your question is yes.
BTW, note that the largest square contained in an n-gon is not always an inscribed square.
For n=5 and n=9, the largest square shares a vertex with the outer polygon, and there are only 3 points of contact with the outer polygon.
Addendum
If n is divisible by 4, it is evident that the maximal inscribed square shares all 4 vertices with the outer polygon P. By inspection, it is obvious that the square has edges in parallel with P if n = 4, 12, 20, ... and not if n = 8, 16, 24, ...
So the complete answer to your question is this:
A maximal inscribed square in a regular n-gon P has edges in parallel with P if and only if n is not divisible by 8.
