Genji
Genji is very patient with you. He tends to take things slow and let you relax while occasionally giving you traditional Japanese remedies and herbal medicines. It seems like he almost acts like a mother when you're sick.
McCree
Jesse will usually cuddle and suffocate you with kisses, even though he'll probably get your illness in return. Sometimes he'll softly sing to you in his relaxing southern voice while rocking you in his arms.
Reaper
Gabriel isn't a man of tenderness, but he'll try his best for you. He'll try to comfort you (usually pats on the back, awkward cuddles, etc.). You find it sweet that he'll let go of his pride and comfort you when you're not feeling well.
Soldier 76
Jack tends to treat you like a little kid. He'll feed you hot soup and make sure that you're resting most of the time. Though you tell him that you're fine, the stubborn man will ignore you and continue on with his inner dad power.
Hanzo
Hanzo tends to give you massages and baths, allowing you to relax. Like Genji, he'll give you Japanese remedies. Hanzo's usually a silent person, but when you're in distress, surprisingly, he'll sing softly in his mother tongue.
Junkrat
Being the crazy clown he is, Jamie tends to be loud and obnoxious, even though he tries his best in comforting you. He isn't experienced in the field of medicine, so he usually buys you random pills and liquid, hoping that it will help (though it usually doesn't).
Sigma
Siebren doesn't stay around you when you're sick. He's too afraid that he'll make your illness worse (somehow). Much like if you're wounded, you'll have to reassure Siebren. And when he does let down his guard, he's as soft as a teddy bear (and as awkward a clown).
Baptiste
Much like McCree, Baptiste will be just as affectionate as he usually is. Though, he does take some precautions, that doesn't stop him from smothering you with kisses.