This answer pieces together the various comments made by Roland Bacher, SJR and gowers above. The proposed generalization of Littlewood's conjecture is false.

As suggested by Roland and SJR, take $r$ to be the golden ratio $(1+\sqrt{5})/2$.
Then $x|rx-y| \le 1$ only when $x$ runs over the Fibonacci numbers $F_k$. Now we want to show that there is an irrational number $s$ such that $F_k s$ is bounded away from integers.

As suggested by gowers there do exist such $s$ for any lacunary sequence $n_k$ (that is a sequence with $n_{k+1}/n_k \ge 1+ c>1$ for all large $k$), and so in particular for the Fibonacci numbers. This is related to a conjecture of Erdos, that for any lacunary sequence $n_k$ there exist irrational numbers $\alpha$ with $n_k \alpha$ not being dense $\mod 1$. Erdos's problem was settled independently by de Mathan and Pollington in a stronger form, showing that there exist many such $\alpha$. We need in fact that the values $\mod 1$ are bounded away from $0$. This is worked out in detail using Pollington's argument in a recent nice preprint of Haynes and Munday (see Lemma 1 of http://arxiv.org/abs/1308.0208 ).

This answer pieces together the various comments made by Roland Bacher, SJR and gowers previously. The proposed generalization of Littlewood's conjecture is false.

As suggested by Roland and SJR, take $r$ to be the golden ratio $(1+\sqrt{5})/2$.
Then $x|rx-y| \le 1$ only when $x$ runs over the Fibonacci numbers $F_k$. Now we want to show that there is an irrational number $s$ such that $F_k s$ is bounded away from integers.

As suggested by gowers there do exist such $s$ for any lacunary sequence $n_k$ (that is a sequence with $n_{k+1}/n_k \ge 1+ c>1$ for all large $k$), and so in particular for the Fibonacci numbers. This is related to a conjecture of Erdos, that for any lacunary sequence $n_k$ there exist irrational numbers $\alpha$ with $n_k \alpha$ not being dense $\mod 1$. Erdos's problem was settled independently by de Mathan and Pollington in a stronger form, showing that there exist many such $\alpha$. We need in fact that the values $\mod 1$ are bounded away from $0$. This is worked out in detail using Pollington's argument in a recent nice preprint of Haynes and Munday (see Lemma 1 of http://arxiv.org/abs/1308.0208 ).