I'm Lucy, and I'd be happy to answer your questions today. I'm sorry for your loss.
If you allow them to move in, they'll become your tenants. Without a lease, they're month-to-month tenants, even if they're not paying rent. What that means is, you can ask them to leave at any time by giving them one month's written notice (30 days in New York City). If they refuse to leave, you'll be able to go to the local landlord/tenant court and evict them. You won't be able to change the locks to force them out, but there's no defense to a properly served 30 day notice. Even though they have kids, they'll be required to leave when the time comes.
A squatter is someone who is living in your property without consent, and squatters don't gain rights until they've lived somewhere without permission for several years, so that's not something you need to worry about.
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