The Fox and Friends crew were chortling over the very idea that the Holy Family as refugees was in the Bible, and made fun of Sharpton, who said nothing wrong.
People sometimes forget that Al Sharpton is actually THE REVEREND Al Sharpton. Secular media personalities might want to think twice before assuming he got something about the bible wrong. That's not to say he's infallible, but he did go to seminary- the Fox and Friends crew probably should have checked their facts before chortling. Of course, Fox and Friends in general is very poor on fact checking. They're sad some horrible things over the years that were completely untrue- not issues of opinion, they were just flat out inaccurate. And we're not talking about like reading the wrong statistics because they were squinting at the teleprompter- we're talking like whole long segments about events that didn't happen.
Fox and Friends is so bad that even Donald Trump watches Morning Joe on MSNBC instead. And when Donald Trump of all people (He who's advisers introduced the phrase "alternative facts" into the popular lexicon) thinks some people on the right-wing are playing too fast and loose with the facts, well, that speaks for itself.
Anyway, to address the question posed by this thread, yes, Jesus absolutely was a refugee in Egypt. He and his family were fleeing their home country to escape persecution from a King who ordered all children of Jesus' age killed. That's almost the definition of refugee. If Mary, Joseph, and Jesus came to a country today seeking asylum, they would immediately be classified as refugees under almost any nation's system. They weren't in Egypt because Joseph was looking for a job or Mary liked the cuisine- they were in Egypt because the Romans wanted to kill their child, who was with them.
I think that situation is very similar to people fleeing from warzones because they think they will be killed for their faith- not just Christians, but also Yazidis, Sufi Muslims, [bless and do not curse][bless and do not curse][bless and do not curse][bless and do not curse]te Muslims, moderate and ethnically Kurdish Sunnis, etc. or killed because they are in the middle of a warzone. Some of the people fleeing were even translators or people who otherwise worked for the US and allied forces and would be tortured or killed by ISIS for that reason if they remained. We have an ethical obligation to keep these people safe by granting them sanctuary until such a time as their countries of origin are safe for them to live in again. We broke it, we bought it.
Now, to be fair, I did notice someone alluding to the idea that the Holy Family's flight to Egypt may be ahistorical. That is to say, there isn't much historical evidence from secular that Herod ever ordered all the male infants under a certain age killed around the time of the birth of Christ- and an event that big would likely be recorded by all kinds of secular historians, we'd find remnants of the scrolls issuing the orders and of contemporary people mentioning losing their children, etc., and that stuff just doesn't seem to exist.
However, it really doesn't matter whether it happened or didn't happen. The story illustrates an important moral truth for us, I think.