They cannot live there for free. The best they could have done was put the payments in escrow and prove they had the money and only with held for some deficiency. They would have to release it to you once remedied. No harm no foul.
Just evict. They are running a scam on you. Unfortunately, once good tenants go bad in the city, they don't get better. EVERYONE has friends/family that educate them on ways to get over in one way or another. EVERYONE has friends/family that has done it before. EVERYONE thinks they are a lawyer and smarter than you until they get served.
If they don't believe you have a valid lease, then they are squatters and still can be sued for eviction.
I grew up in this city. I am very familiar with many of the ways in the city. You should act quickly and get them out. You need to have someone secure the property immediately or coincidentally, your furnace, HW Heater, and copper will come up missing within days of their moving. Start all over with another tenant.