As far as an actual physical illness goes, there is nothing wrong with Betty. What is really going on is that she is scared witless, and acting sick in order to not get in trouble. The night before, her dad, Reverend Parris, discovered her and her friends dancing in the forest. Dancing, in Puritan communities, was strictly forbidden, and warranted a whipping. And, not only that, Abigail and others were casting spells and making charms to do harmful things; Betty knows this, and that is even worse than dancing. That is indicative of witchcraft, whose punishment is hanging. So, when her dad jumps out and discovers her in the middle of all of this mischief, she gets so scared that she nearly passes out. She doesn't want to get whipped or hanged, so, she pretends to be ill instead.
If she is "ill," she can't answer any questions about what they were doing that night, and her father will put aside her punishment because he is so worried about her. So, she pretends to be sick; she refuses to eat or speak, and lies there like she is in a coma. Occasionally, she'll break out in hysterics, flying at the window to try to "fly." This bizarre behavior works well to distract her father; concerned, he sends for a doctor, and then for Reverend Hale, just in case witchcraft is involved. As Abigail tells Proctor in Act One, Betty isn't sick, "she took fright, is all."
I hope that those thoughts help a bit; good luck!