Can I be cleared out of the barracks and not receive BAH?

If you're being kicked out the barracks your unit must provide you with a certificate of non availability that authorizes you BAH. That's not granted by the unit, but the post commander.

Well, this is a odd one. I was a little confused by "They want us to clear Tuesday". Who is us and how many?

Generally they can't kick you out if you rate a room and you certainly do. But there is more than likely more to this than we as the RP community know. Who is kicking you out of the Barracks? Why are they doing it? Did something happen? Is the barracks at capacity? I'm doubtful anyone of substantial rank knows or they would have already squashed it and probably would have sat you down about living out in town out of pocket. This is the kind of stuff that send E-7s and above crawling up the walls. Eventually this is going to boil over and cause a stink. Either you bring it up the chain and it gets fixed which also will result in a chewing out for moving out in town when you don't rate BAH. Or you keep it quiet, they remove you from the barracks whoever they are and eventually it comes out you have been living on post at your own expense and both you and whoever kicked you out of the barracks get a chewing out.

The choice is yours but this whole scenario is born from riding under the radar trying not to skyline yourself.

I do feel that some details may be missing in this story. Who is making you clear the barracks? For what reason do you have to clear the barracks? If you maintain that room, you cannot be kicked out of it. Have you done something to cause this to happen? Are you the only one losing their room? It is odd to me that they would randomly choose one soldier to lose their room and it just happens to be a soldier who is not even using it.
Not saying you're wrong in this situation, it is just hard to come up with any reasonable thought, when so many details are missing. However, yes you will be entitled to BAH if they state that there are not enough rooms available to accommodate you. However, the list usually starts with the most senior members in the barracks, ie: Sergeants.