First we show that it is possible for all $n$ odd. Say a block is a set of consecutive beads of the same color such that the beads immediately to its left and right are both the opposite color. Inductively, it is easy to see that a block of odd length can, in a sequence of moves, be changed completely to the opposite color. As long as there are still beads of both colors, some block must have odd length (keeping in mind that $n$ is odd), so we can eliminate it and strictly decrease the number of blocks. Thus it is clear that continuing in this manner we will eventually reach a monochromatic necklace. Here are constructions that show even $n$ fail: for $n \equiv 0 \pmod{4}$ we partition the necklace into sets of four consecutive beads and color each one $BBRR$; no moves can be made at all. For $n \equiv 2 \pmod{4}$ we pick two adjacent beads to color $BB$, and then partition the rest of the necklace into sets of four consecutive beads and color each of those $BBRR$. Verify that after two moves we must return to an isomorphic configuration, meaning we will never reach a monochromatic necklace.