What does it imply when someone is called “Lucifer in the flesh”?
Solution 1:
As you understand, Lucifer means the same thing as Satan, considered in Christianity to be the leader of evil spirits.
The word flesh sometimes has a sexual connotation, but not in this case. Here, it's part of an expression in the flesh, which the Cambridge English Dictionary defines as meaning "physically in front of you," with the example sentence "I’ve seen her perform on television, but never in the flesh." It's usually synonymous with the expression "in person."
In this case, since it is applied to an entity that is normally considered a spiritual being, it might also have a more literal meaning: not just "Lucifer, present in person" but "Lucifer in a physical human body." A synonym in this sense would be incarnate: we can say "Lucifer incarnate" to describe a person as evil.
It seems to me to have a strong, but vague meaning: "a very evil/bad/terrible person." I don't think the phrase carries any more specific connotations. In a religious context, the phrase might be used literally, but in most other cases it will just be a hyperbolic insult.
Solution 2:
It means he's as evil as The Devil in human form.
Solution 3:
To be more specific, (and I think Boehner meant it in this sense):
Cruz's reputation, fair or unfair, is for pride— acting as if he was 'above' others that he was working with.
In the Bible, Lucifer (which is a later Latin name applied to that entity) was expelled from heaven because he placed himself 'above' God.