With nothing in the structure "house", it is almost impossible for the structure to be a "home", because a house (structure) has to be a place you can live, has to have something in it to enable you to dwell in it - although there are exceptional circumstances where people call places "home" which offer nothing or next to nothing to support dwelling there.
I think he is referring to the Russian, as in Russian when you refer to nothing, never, nowhere, etc. you have to negate the verb (use a doubke negative). So in english we say "I never read." but in Russian we say "Я никогда не читаю." Which literally translates as "I never not read."
If your definition of home "a house with nothing in it", then do I no longer have a home if someone took everything out of it? They didn't pick up the house and put it in their pickup truck.
The homeless squatters' homes are homes because that's where they choose to stay. If I choose to live in the sewers for that night, that's my home. The property (no pun intended) of if something is a home should be if the one who calls it home... calls it home.
We should also note the peculiarity of both interpretations. It is inherently paradoxical to say that a "homeless squatter" has a home (even if they brought a belonging or three to qualify as you define it) has a home, because then they're no longer homeless by definition. Then nobody in the world is homeless unless they spend every waking second roaming the world.
Houseless just doesn't sound the same. Perhaps there's an emotional requirement to something being a home too?
I hope you see now how ridiculous it is to reduce that difference down to being a covered shelter with stuff in it.
Another poster mentioned somewhere else in the comments for this question that ''house'' pertains to the structure itself while ''home'' implies emotional attachment, typically from a length of time spent living and building memories within the structure.
As for your assertion that ''homeless squatter'' is an impossibility, your definition of homeless is fairly narrow. The barest qualification for being homeless is to lack a permanent address. Therefore, if someone is living in an abandoned house and squatting on that property, that person is still homeless because the house does not legally belong to him or her. An individual can live in a hotel room, a vehicle, or on a friend's couch and be just as homeless as someone taking shelter in a tent community, under a bridge, or sleeping on a park bench. Homelessness is not measured by type of structure, possessions, income (because it's wholly possible to work and still be homeless!), or even appearance. It's just the status of not having a permanent residence.
Interesting question, because so far, I've only encountered people or animals "having" things, in the у [genitive pronoun] есть or у [genitive pronoun] нет formats. If things cannot "have" other things (у дома есть... or у дома нет... are invalid), then your suggested answer might be correct, because there isn't another way of saying that.
However, I think that "В этом дома" = "in this house" is key to translating this sentence, and has to be included in the answer, which precludes use of "This house has".
While the intent of your translation is more-or-less the same idea, it seems to me that it's too far away from the literal translation to be valid as an answer to this exercise. Artful translation might indulge you in the license you take with the wording, but it's not sufficiently accurate for a language course.