Some bases, the BOQ's and BEQ's are still separate, and you're right, the "CBQ" is just a joint front desk for the two. Some newer Q's combine them for E-7+ and it's really more "Bachelor Khaki Quarters". Some will extend that down to E-6 or even E-5, depending on demand. I think it really depends on demand for the rooms; places like P'cola or Norfolk, that have a lot of transient Officer students and/or Officer Reservists, seem to keep the Other Ranks in their own digs (mostly, I'd guess, so that the junior guys don't get squeezed out and have to stay out in town on a smaller paycheck).
I have no idea how or when that changed, or even if it's really Navy-wide policy or just whatever individual bases do to keep the rooms filled. I do know that I've stayed in BOQ's in Coronado, Point Mugu, Jax and Pearl Harbor and seen Chiefs and First Class staying in rooms just down the hall from me.
Back to the OP's question: whenever possible and for the duration of your Naval career, stay out in town. The ONLY place I've ever been where the Q was preferable to In Town was Brunswick, ME, for SERE. They sent you to this bizarre shithole of a fleabag, and the Q rooms were actually pretty nice.