It follows from the irrationality of $\pi$ and [Weyl's criterion][1] that the positive integers are equidistributed modulo $\pi$. In particular, asymptotically one-third of the positive integers $k\in\{1,\dotsc,n-1\}$ are such that $k\bmod\pi$ lies in $[\pi/3,2\pi/3]$. It follows, for $n$ large, that
$\prod_{k=1}^{n-1}\cos^2(k)\leq 2^{-n/2}$. Hence the limit in the original question equals zero.

  [1]: https://en.wikipedia.org/wiki/Equidistributed_sequence#Weyl's_criterion