which values of n do a non-cyclic Gray code exists
Standard binary Gray codes are powers of two and have the cyclic property in all cases.
There are likely an infinite number of non-cyclic Grey codes so I'm not sure your question can be answered completely. For a specific example consider the Gray code sequence for 7:
011, 010, 000, 001, 101, 100, 110
The ends of the sequence differ by more than a single bit change so it is non-cyclical.