People are saying that if you're going to let immigrants in you must be willing to open your house to them.

With the caravan of immigrants coming north, many who are against this are using one argument in particular. They are saying that if we let them in "you should open your house to house 3 or 4 of "them."" I find that it's a minor reason to not allow people to escape injustice. It's as if it's a false equivalence or a scare tactic. Where they end up seems irrelevant plus what if their next door neighbor becomes a sponsor family. I know there's a fallacious argument in there. Thank you.

You can use an analogy to show how nonsensical this is. You can ask, "If we allow people to give up children for adoption in the United States, then must be willing to adopt 3 or four children? If not, why the difference?" I don't think there is a fallacy per se, but I do think an unreasonable claim is being made.

I guess that's a non-sequitur. Maybe a response to that would be "If you let certain people move in to your neighborhood, does that entail that you have to accomodate some of them in your house?", and since this obviously is ridiculous and the answer would be no, follow up with "Alright then how does letting certain people inside the borders of the entire country necessitate that you let some of them inside your house?"

Well you started out with a straw man...it isn't letting immigrants into the country...it's ILLEGAL immigrants. It's illegal, period. No debate about that.....

This question does not correctly state the issue:
"They are saying that if we let them in 'you should open your house to house 3 or 4 of 'them.' "

The correct issue is different and could be stated as follows: "If you demand that the Caravan people be admitted despite my deep concerns for the potential financial costs to me as well as in terms of the safety and security of myself and my family, you should be willing to ensure that you take on your fair share of the possible costs and consequences by opening the gates of your gated community to immigration and directly sponsoring 3-4 members of the caravan."

You might respond, "That's silly. We all pay taxes, and will equally share the burden of supporting the new arrivals until they get established and become contributing members of our society." In the end, we will all be better off. "Besides that, it is the just, and fair, and right thing to do."

Let's look at the utilitarian argument more deeply via a cost/benefit analysis--in particular, Cui Bono, who benefits? In this situation, obviously the migrants would benefit. In addition, owners of large farms and factories would benefit from the availability of cheap labor. Wealthy people who live on islands, on mountain tops, and inside gated estates and protected communities would benefit from the cheap labor on their estates, and from enhanced financial portfolios. The average citizen would experience maximal levels of life satisfaction and benefit from being just, virtuous, and righteous and would undoubtedly be admitted directly to Heaven if there were such a place. But what about the costs? The taxpaying public would have to pay for the schools, healthcare, legal and welfare benefits for the new arrivals-- and even some additional law enforcement since we have all of these restrictive laws that a few of the migrants seem to be ignoring.
For the sake of argument, let's assume that the cost of supporting the migrants is $1000 per year for each taxpayer. Mr Smith earns 50 thousand dollars per year. Ms. Movie Star earns an average for 20 million dollars per year. That 1000 dollars means Mr. Smith's kids will have to forego dental care for the year. That 1000 dollars is what Ms. Movie Star usually tips her hairdresser at Christmas time.
Since there are many more Smith families that Ms. Movie Stars, I doubt the utilitarian argument would hold up, and we would not achieve the "best for the most" by admitting the Caravan. If this is correct, we then have to revert to arguments based on altruism, duty, justice, equality or fairness. But those are for another day.

Strikes me as a variation of Slippery Slope (also known as absurd extrapolation, thin edge of the wedge, camel's nose, domino fallacy)

Description: When a relatively insignificant first event is suggested to lead to a more significant event, which in turn leads to a more significant event, and so on, until some ultimate, significant event is reached, where the connection of each event is not only unwarranted but with each step it becomes more and more improbable. Many events are usually present in this fallacy, but only two are actually required -- usually connected by “the next thing you know...”