Yes, but Joseph, Mary, and Jesus went back home again as soon as the threat to Jesus' life was over, and they did not receive public assistance while they were there.
2 Thessalonians 3:7-9 For you yourselves know how you ought to imitate us; we were not idle when we were with you, 8 and we did not eat anyone’s bread without paying for it; but with toil and labor we worked night and day, so that we might not burden any of you.
Also, when God commands a person to immediately move his family to Egypt, it is best to obey immediately.
I do not see a correlation between the Holy Family's flight to Egypt and the horde of millions of illegals invading our country.
According to the ancient traditions of the Coptic Church (the indigenous Christians of Egypt since the time St. Mark the Evangelist established a base of apostolic and missionary operations in Alexandria, turning Alexandria into one of the great early centers of Christianity) the Holy Family in their travels to and within Egypt, experienced many things--including being welcomed, cared for, fed, and provided welfare by locals. Where they were safe from the threats and dangers of Herod--because of God's Providence operating in and through Egyptian locals.
It's not as though the Holy Family fled to Egypt and then hid in a cave all alone until they returned--they would have, even if some of the traditions and stories contain elaborations and legendary accretions (such as the Child Jesus hallowing out a spring by His Divine power), it would be pretty obvious that for a family with a small Child they would rely on the kindness, compassion, and hospitality of strangers in the land of their refuge.
As for the returning part. Yes, the Holy Family did return to Palestine after their refuge; even as many who cross the southern border come to work for a short period of time, and send money back home, and often return home themselves. Sure, lots of people cross, and stay--to make a better life for themselves and their family in a new place. But many are also temporary workers, they don't stay, but they work and provide, and then return.
Perhaps a better comparison might be the Exodus? Where according to the Bible up to a million people, Hebrews and a "mixed multitude" with them, fled Egypt following Moses into the desert of Sinai, where they received God's Torah, becoming the Jewish people; they--or rather their children--would after forty years in the wilderness enter into Canaan. And, at least according to the Bible, that was an invasion. Perhaps you recall how God ordered the Israelites to circle Jericho and the walls came tumbling down. Or perhaps you recall the conflicts with the locals, such as the Amalakites, who had harassed and assaulted the Israelites while in the Negev just past Sinai, and eventually resulting in the somewhat infamous quote in Scripture where the Amalakites are to be utterly destroyed and erased from under heaven.
While I am speaking entirely tongue-in-cheek here, given what I often hear from many, America is practicing a multitude of abominations, with comparisons made to Egypt, Canaanites (often a comparison is made between abortion and the cult of Molech), and Babylon; so perhaps if there is an invasion from the southern border, who is to say it isn't the Divine Will of God who is putting America under judgment for her multitude of abominations, isn't it written,
"
You are the daughter of your mother, who loathed her husband and her children; and you are the sister of your sisters, who loathed their husbands and their children. Your mother was a Hittite, and your father and Amorite. And your elder sister is Samaria, who lived with her daughters. Not only did you walk in their ways and do according to their abominations; within a very little time you were more corrupt than they in all your ways. As I live, declares the Lord God, your sister Sodom and her daughters have not done as you and your daughters have done. Behold, this was the guilt of your sister Sodom; she and her daughters had pride, excess of food, and prosperous ease, but did not aid the poor and needy. They were arrogant and did an abomination before Me. So I removed them, when I saw it. Samaria has not committed half of your sins. You have committed more abominations than they, and have made your sisters appear righteous by all the abominations that you have committed. Bear your disgrace, you also, for you have intervened on behalf of your sisters. Because of your sins, in which you acted more abominably than they, they are more in the right than you. So be ashamed, you also, and bear your disgrace, for you have made your sisters appear righteous." - Ezekiel 16:45-52
We could go full-biblical: America is an intoxicated harlot drunk on wealth and power, greater empires than she have been taken out for less. Perhaps, if there is an invasion, it is because America has been an unrighteous hell-bent empire that has grown fat on consumption, for her great mistreatment of the poor and hungry in her midst, because her leaders have grown fat on the abuse of the weak and the needy, and the day of her judgment has finally drawn near.
Do I actually believe that? Well no. But, if we are going full-biblical when talking about modern day America...
-CryptoLutheran