I'm getting Med Boarded and going on terminal leave soon. Do I still collect BAH while on leave?

Not only is one form of leave functionally no different than another, but you're not separated. If you're entitled to BAH, you remain entitled to BAH until your date of separation. I'd go back and address this with the manager, and if they're going to continue that BS, I'd go to your commander or first sergeant.

Technically true, though they've got the reasoning backwards. Once you move out, HOUSING is no longer able to get your BAH. Local policies will dictate whether you can stay in housing on transition leave, though I haven't seen one where you're required - not asked - to leave. Thing is... as soon as you leave, they'll clean the house and re-rent to someone with more longevity... a business decision on their end.

YOU, on the other hand, will continue to receive BAH through the end of your transition leave, regardless of your lodging arrangements. The only question is whether there will be a matching deduction going to privatized housing.

Same answer as MAJ Jennifer H. This sounds like a local policy. Go to the manager of the group, if you are not separated you are entitled to BAH when on terminal leave.
If you requested to sell your remaining leave days, only then you would get base pay w/o BAS,BAH.