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.

Does your unit know you live off post? If not, there's a butt chewing coming your way for that. Did your 1SG explain why you have to clear the barracks? Did anyone explain? Normally, if they are removing you from the barracks, they need to find you new barracks space. If none are available, then a statement of non availability is to be provided.

When I was a young troop (yeah, that was a LONG time ago), one could get the commander's permission to reside off-post at his own expense. BAH would not be authorized, but the commander could approve BAS. If adequate quarters are not available in the unit barracks, the billeting office can provides certificates of nonavailability, normally granted to more senior personnel first.

EDIT most likely this is because of over population and the barracks are full. But if that is the case. If I'm not mistaken I should be signing paperwork to receive BAH before I sign the room to be cleared

I'm wondering if the OP was required to get permission to move off post. I ask because I had to get permission to attend college night courses while I was attending a school for my reclassification into another MOS.

It might depend on the reason for asking you to clear out. If they are hurting for space, that centers around occupancy rate. This could trigger you being approved for BAH, due to a lack of billeting. It's up to Base Billeting to determine this, and your CoC to implement. We always required brown baggers to maintain a rack, same as you speak of. Sgts and above got first shot at BAQ (BAH), and case by case after that. They may have operational concerns due to your job requiring close proximity to base. Unless it is based on your conduct, it is limited to providing either a rack or BAH, so use your CoC to get it resolved.