No it is not correct. You should ask "Do you feel unwell?". You might hear people say something, like what Brian gave in his answer. For example: "I feel unwell. Do you feel unwell, yourself?" - note the comma. here, in this sentence, by "Do you feel unwell, yourself" they mean "do you also feel unwell". it is common spoken English, not formal English. Examples of sentences with feel + yourself; Do you feel confident in yourself. Do you feel happy with yourself. Are you comfortable in / with yourself.

More or less correct, with slight modifications, as in the following context: Paul: I'm feeling a little sick today. I have a sore throat. Peter: I'm sorry to hear that. I think I also have a scratchy throat. Paul: "Are you feeling unwell yourself?" Note I changed "do you feel unwell yourself," which is a general, non-time-specific comment, to "are you feeling unwell yourself," which specifically means one is feeling sick "now." The use of "do you" makes this question an open-ended, non-time-specific question, and unless someone who is chronically ill feels unwell on a regular basis, it is better to use "are you feeling" rather than "do you feel." Also the addition of "yourself" makes this question imply that the asker also feels unwell.

"Are you feeling unwell yourself?" as a reply is more natural.