Numbers with anticlosed sets of non-squares

[jens at voss-ahrensburg.de]

Let R be a ring. We will call a subset T of R *anticlosed* if for any
elements t1 and t2 of T, neither the sum t1 + t2 nor the product t1 * t2
lies in T.

For an integer n >= 2, let R := Z/nZ, and let T be the set of non-squares
of R. For which n is this set T anticlosed?

I ran a quick search with the computer and found that within the first
10000 integers, only 2, 3, 5, 6, 7 and 10 have this property, so it
appears that this definition may yield a finite (and quite short)
sequence. Any idea on how to prove that no numbers beyond 10 have an
anticlosed set of non-squares?

Regards,
Jens