Is it considered harmful for all three sons in a family to study medicine due to the evil eye, by analogy with the noble verse: {And he said, "O my sons, do not enter from one gate but enter from different gates"}?

There is no objection to brothers specializing in medicine or other beneficial sciences. One should not abandon good deeds out of fear of the evil eye. It is preferable to take preventative measures against the evil eye, such as seeking protection through prophetic remembrances (adhkar) and supplications (du'as), and following the example of Allah's saying: {And he said, "O my sons, do not enter from one gate but enter from different gates."} Some exegetes have stated that Jacob forbade them from entering from one gate out of fear of the evil eye due to their strength and beauty, yet he informed them that it would not avail them anything against Allah, and that he relied upon his Lord. There are other interpretations of the verse. And Allah knows best.