I've been told that somewhere, there is an option specific to Soldier 76 (and a few other specific classes) that allow you to see teammates health. For characters like Mercy it is enabled by default, but for Soldier 76 you have to enable it.
I can't figure out where to go to do that though.
Best Answer
If you go into the options menu, you'll see the option to toggle teammate health. Note that Zarya also has this option as well; the exact instructions that follow are the same except to pick Zarya from the All Heroes dropdown menu.
As of an older update Blizzard made it able to enable this option across all heroes.
You'll need to go to:
The option is labelled "Allied Health Bars". Toggle On to show.