Whether or not they're paying their mortgage is irrelevant. If you stop paying your rent, you are in breach of your lease and could be evicted for non payment of rent. And you do NOT want an eviction on your civil record.
If you have a lease with your landlord, make sure you keep a copy or request a copy if you don't currently have one. Once the bank takes possession, it's possible they might offer you 'cash for keys' to help you move. But if you have no proof that you had a lease, they might not offer you this incentive to move.