控件主要代码

///倒计时按钮
class CountDownButton extends StatefulWidget {
  Function countDownPressed;
  bool countDownStatus;

  CountDownButton(
      {Key key, this.countDownPressed, this.countDownStatus = false})
      : super(key: key);

  @override
  State<StatefulWidget> createState() {
    return _CountDownButton();
  }
}

class _CountDownButton extends State<CountDownButton> {
  int countDowmTime = 60;
  String countDownText = '发送验证码';

  @override
  Widget build(BuildContext context) {
    if (widget.countDownStatus && countDownTimer == null) {
      countDownTimer = Timer.periodic(Duration(seconds: 1), (timer) {
        setState(() {
          if (countDowmTime - timer.tick > 0) {
            countDownText = '${countDowmTime - timer.tick}秒';
          } else {
            countDownText = '发送验证码';
            widget.countDownStatus = false;
            countDownTimer.cancel();
            countDownTimer = null;
          }
        });
      });
    }
    return OutlineButton(
      onPressed: widget.countDownStatus ? null : widget.countDownPressed,
      child: Text(countDownText),
    );
  }

  Timer countDownTimer;

  @override
  void dispose() {
    countDownTimer?.cancel();
    countDownTimer = null;
    super.dispose();
  }
}

主要思路

Flutter中的控件主要分为有状态的和无状态的，在这里，主要使用的是有状态的控件进行实现。
CountDownButton类中的构造函数中自定义了两个参数countDownPressed（按钮点击后响应的方法）以及countDownStatus（当前按钮的状态：可发送和倒计时中）。
countDownStatus为true的时候，在build方法中则开启一个周期性的定时器，每秒触发一次，修改按钮中的文本

使用实例

class RegisterPage extends StatefulWidget {
  @override
  State<StatefulWidget> createState() {
    return _RegisterPage();
  }
}

class _RegisterPage extends State<RegisterPage> {
  BuildContext _context;
  bool sendCodeStatus = false;
  TextEditingController accountController = TextEditingController();
  TextEditingController codeController = TextEditingController();
  TextEditingController passwordController = TextEditingController();
  TextEditingController passwordAgainController = TextEditingController();

  @override
  Widget build(BuildContext context) {
    _context = context;
    return Scaffold(
      appBar: AppBar(
        title: Text("Register"),
        centerTitle: true,
      ),
      body: Container(
        padding: EdgeInsets.fromLTRB(10.0, 10.0, 10.0, 0.0),
        child: Column(
          children: <Widget>[
            Row(
              children: <Widget>[
                Text("账号"),
                SizedBox(
                  width: 10.0,
                ),
                Expanded(
                  child: TextField(
                    obscureText: false,
                    controller: accountController,
                    decoration: InputDecoration(
                      hintText: '请输入账号',
                    ),
                  ),
                )
              ],
            ),
            Row(
              children: <Widget>[
                Text("验证码"),
                SizedBox(
                  width: 10.0,
                ),
                Expanded(
                  child: TextField(
                    obscureText: false,
                    controller: codeController,
                    decoration: InputDecoration(
                      hintText: '请输入验证码',
                    ),
                  ),
                ),
                CountDownButton(
                  countDownStatus: sendCodeStatus,
                  countDownPressed: sendCodeClick,
                ),
              ],
            ),
            Row(
              children: <Widget>[
                Text('密码'),
                SizedBox(
                  width: 10.0,
                ),
                Expanded(
                  child: TextField(
                    obscureText: true,
                    controller: passwordController,
                    decoration: InputDecoration(
                      hintText: '请输入密码',
                    ),
                  ),
                )
              ],
            ),
            Row(
              children: <Widget>[
                Text('确认密码'),
                SizedBox(
                  width: 10.0,
                ),
                Expanded(
                  child: TextField(
                    obscureText: true,
                    controller: passwordAgainController,
                    decoration: InputDecoration(
                      hintText: '请输入密码',
                    ),
                  ),
                )
              ],
            ),
            Row(
              children: <Widget>[
                Expanded(
                  child: RaisedButton(
                    onPressed: registerClick,
                    child: Text('确认'),
                  ),
                )
              ],
            )
          ],
        ),
      ),
    );
  }

  void sendCodeClick() {
      Toast.show(_context, msg: '发送验证码成功');
    setState(() {
      sendCodeStatus = true;
    });
  }