My brother signed a lease that says if they are 10 days late with rent they can be asled to move. Can he & his family be kicked out?

ANY question on law needs to include your general locale as laws vary by area.

That is actually very generous. In some places if they are 1 day late they can be given 3 days to pay up or vacate. The landlord still has to go to court to obtain a legal eviction order.