Align Text widgets in a row widget

IntrinsicHeight is the widget you are looking for. It's working fine with that widget.

@override
  Widget build(BuildContext context) {
    return Scaffold(
      body: new IntrinsicHeight(
        child: Row(
        crossAxisAlignment: CrossAxisAlignment.start,
        children: <Widget>[
          new Align(
            alignment: Alignment.topLeft,
            child: Container(
              padding: const EdgeInsets.symmetric(horizontal: 10.0),
              child: Text('"'),
            ),
          ),
          Expanded(
            child: Text(
              "The middle text entry can span several lines, but the first two will be really small. I'm wanting to have the first widget have the text at the top of the row and the 3rd widget to have the text at the bottom of the row. It's basically something similar to this image"
            ),
          ),
          new Align(
            alignment: Alignment.bottomRight,
            child: Container(
              padding: const EdgeInsets.symmetric(horizontal: 10.0),
              child: Text(
                '”',
                textAlign: TextAlign.start,
              ),
            ),
          ),
        ],
        ),
      ));
  }

You can always do Column -> [Expanded(text), Expanded(Container(), Expanded(text)] and adjust the flex of the blank box as needed.

There's also Container (or Column/Row) -> Stack -> [Column(mainAxis start), Column(mainAxis end)].

Work smarter, not harder. IntrinsicHeight widget is computationally expensive and I would not recommend using it for this.