Your job is to find bugs that could negatively impact those using the program, and in the health arena, these bugs could also potentially impact patients as well.
If you find flaws and true bugs that need to be addressed, you should flag them, end of the story.
My experience with testing comes from beta testing games, which I know is a less vital task, but the end result is the same. The more things that are corrected before the launch of the product, the better the initial sales will be and the better the reception of the product will be.
I wouldn't worry too much about how the coders respond to you. They do work hard at what they do, and it is a blow to the ego/pride to have someone find flaws in your work (again, personal experience). But as long as you are not rubbing these flaws in their faces, or acting high and mighty about things, any friction should pass.
In the end, everyone just needs to work at their given functions as best as they can to ensure that the product released is in the best possible state.
And don't worry, the general public will find bugs that even you missed. (personal experience again, lol)