A square in the plane with 4 vertices of the same color

I believe the answer is

Not a proof, but a heuristic argument:

EDIT: A "proof" by computer. At least this gives the evidence for the yes/no part, so that a more serious attempt can be made in the right direction.

Not an answer, but a suggestive result.

That doesn't prove that a square is inevitable, because it leaves open the possibility that

There is a relevant paper which I came across recently connected to this question:

• "Extremal binary matrices without constant 2-squares" by Roland Bacher and Shalom Eliahou

In it, the authors prove that

which shows not only that

but also that

Further interesting points