How to set the background color of a Flutter OutlineButton?

To modify the backgroundColor of a OutlineButton you can use a DecoratedBox and a Theme widget. At the end of this answer you'll find a quick example.

Anyway I'd still recommend simply using the FlatButton with its color attribute instead.

Wrap your OutlinedButton inside a DecoratedBox. Set the shape of your DecoratedBox to the same shape your OutlinedButton. Now you can use the color attribute of your DecoratedBox to change the color. The result will still have a small padding around the OutlinedButton. To remove this you can wrap the DecoratedBox inside a Theme in which you adjust the ButtonTheme. Inside the ButtonTheme you want to set materialTapTargetSize: MaterialTapTargetSize.shrinkWrap.

The padding is added inside Flutter, to increase the tap area around the button to a minimum size of 48x48 (source). Setting materialTapTargetSize to MaterialTapTargetSize.shrinkWrap removes this minimum size.

---

FlatButton example:

import 'package:flutter/material.dart';

void main() => runApp(MyApp());

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      home: Scaffold(
        body: Center(
          child: FlatButton(
            color: Colors.pinkAccent,
            shape: CircleBorder(),
            onPressed: () => {},
            child: Text('A'),
          ),
        ),
      ),
    );
  }
}

---

OutlinedButton example:

import 'package:flutter/material.dart';

void main() => runApp(MyApp());

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      home: Scaffold(
        body: Center(
          child: MyButton(),
        ),
      ),
    );
  }
}

class MyButton extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return DecoratedBox(
      decoration:
          ShapeDecoration(shape: CircleBorder(), color: Colors.pinkAccent),
      child: Theme(
        data: Theme.of(context).copyWith(
            buttonTheme: ButtonTheme.of(context).copyWith(
                materialTapTargetSize: MaterialTapTargetSize.shrinkWrap)),
        child: OutlineButton(
          shape: CircleBorder(),
          child: Text('A'),
          onPressed: () => {},
        ),
      ),
    );
  }
}

It is pretty much easy, just use

backgroundColor: MaterialStateProperty.all(Colors.colorname),

OutlineButton has been deprecated and has been replaced with OutlinedButton. (Notice the "d".)

OutlinedButton's documentation helped me understand how to use it. Here's an example with these states: Disabled (Grey) --> Normal (Blue) --> Pressed (Red)

   return Container(
      width: double.infinity,
      height: 48,
      child: OutlinedButton(
        child: Text(
          "This is an Outline\"d\"Button. (Not OutlineButton)",
          style: TextStyle(color: Colors.white),
        ),
        onPressed: () => print("Tapped"),
        //onPressed: null, //Uncomment this statement to check disabled state.
        style: ButtonStyle(
          backgroundColor: MaterialStateProperty.resolveWith<Color>((states) {
            if (states.contains(MaterialState.disabled)) {
              return Colors.grey[100];
            }
            return Colors.blue;
          }),
          overlayColor: MaterialStateProperty.resolveWith<Color>((states) {
            if (states.contains(MaterialState.pressed)) {
              return Colors.red;
            }
            return Colors.transparent;
          }),
          side: MaterialStateProperty.resolveWith((states) {
            Color _borderColor;

            if (states.contains(MaterialState.disabled)) {
              _borderColor = Colors.greenAccent;
            } else if (states.contains(MaterialState.pressed)) {
              _borderColor = Colors.yellow;
            } else {
              _borderColor = Colors.pinkAccent;
            }

            return BorderSide(color: _borderColor, width: 5);
          }),
          shape: MaterialStateProperty.resolveWith<OutlinedBorder>((_) {
            return RoundedRectangleBorder(borderRadius: BorderRadius.circular(16));
          }),
        ),
      ),
    );

Flutter replaced the former 3 button types (FlatButton, RaisedButton and OutlineButton) with new buttons types (TextButton, ElevatedButton and OutlinedButton) to remain in sync with Material design and also because using MaterialStateProperty provides the ultimate flexibility to achieve whatever state-specific-UI one needs. You can read more about it here.