When people don't have to pick their housing based on the location of low-wage jobs (dense cities), I would expect the rent to go down due to better utilization of the total housing supply.
With BI, you increase the purchase power of a whole bunch of poor people. People that didn't live in the city before, now think they have a shot and can afford rent for a while and get a job in the city!
Demand goes up, prices go up. If you still want to live in the city, you will need to spend all of your BI-income AND all of your pay on the VERY SAME basic necessities.
Sounds pretty obvious to me and I can't really see it turn out any other way.
Moving back to the country side, where it's cheap ? Ok, you can live from BI, but you still don't have a job, your kids don't get a proper education and their future looks just like yours. Barely getting by and sitting around watching tv.