No, I don't think using "should" would make much sense here, so I think you ought to stick with "would". Perhaps I can make this clearer by rearranging the sentence a little (just for the purpose of making the meaning clearer, I'm not suggesting you rearrange it in general use):
You are feeling ill. If I were you, I would stay at home.
In other words, you are saying that if you were in the ill person's place, you would choose to stay at home. The alternative would be
You are feeling ill. If I were you, I should stay at home.
Here the "I should" construction is confusing and doesn't really hold much meaning, as the "if" clause is already making the sentence conditional. "I would" gives a definitive statement of what you would do if you were in that person's situation.