Not really a cliche, quite a lot of times when you have a fever you are more red in the face.
No, it's definitely a cliche. The red flush of a fever is distinctly different from someone who's blushing out of embarrassment, and I'm pretty sure feverish people aren't leaping around the room in an energised frenzy whilst stuttering out denials to inquiries into their health.
No idea how anybody can think a blush is anything like a fever. A fever is a symptom of an illness, for one, and it's very rarely the only one.