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Let $S=\{00,11\}$ and $a=10$. The largest antichain in $S$ has size 1, while the largest antichain in $S_a$ has size 2. We can extend this construction to get an affine subspace of any dimension $k$, all of whose elements form an antichain (of size $2^k$). Namely, letting $0^j$ denote a string of $j$ $0$'s, let $S$ be generated by $110^{2k-2}$, $00110^{2k-4}$, $0^4110^{2k-6}, \dots, 0^{2k-2}11$, and let $a=101010\cdots 10$.