The important thing is to use the same form for each phrase. In the first case, I think your 1st phrase is clearer as it highlights both illnesses, but the 2nd phrase is also acceptable.
In your second paragraph, the 2nd example you give is better ("They are looking for a cure for both the Zika and the flu virus") for the reason above - keep the form parallel. It just sounds better.
Keep up the good work. Not many Portuguese speakers have a good facility in English.