Control of individual Framed edges?

Would a CellFrame around a TextCell work?

Framed[
 x -> TextCell[
   Column[{"Depression", "PTSD", "Diabetes Type II", "Smoker"}], 
   "Text",
   CellFrame -> {{True, False}, {False, False}}]]

An alternative way of getting a similar display would be to define a new function leftFramed that only puts a vertical extensible line to the left of the content:

leftFrame /: MakeBoxes[leftFrame[obj_], _] := 
 RowBox[{"\[LeftBracketingBar]", ToBoxes[obj]}]

Framed[x -> 
  leftFrame[
   Column[{"Depression", "PTSD", "Diabetes Type II", "Smoker"}]]]

You can see that this vertical line is in fact an extensible character, not a graphical line.

In the same vein, you could also use other extensible symbols to decorate the left side:

leftBrace /: MakeBoxes[leftBrace[obj_], _] := 
 StyleBox[RowBox[{"{", ToBoxes[obj]}], SpanMaxSize -> Infinity]

Framed[x -> 
  leftBrace[
   Column[{"Depression", "PTSD", "Diabetes Type II", "Smoker"}]]]