Numbers with anticlosed sets of non-squares

[hv at crypt.org]

jens at voss-ahrensburg.de wrote:
: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?

Am I misunderstanding something or is '7' a typo for '9'? I get:
  n = 7 => T = { 3, 5, 6 } and 3 + 3 = 6
  n = 9 => T = { 2, 5, 8 }, and [ T, + ] = [ T, * ] = { 1, 4, 7 }

Hugo