首先，这篇文章在CodeReview中会更好.

First of all, this post would have fitted better in CodeReview.

第二，我可以想象，你确实想做什么. 为简化起见，我建议您不要针对您的情况使用通用集合.

Second, i can imagine, what you did want to do. To shorten things, i recommend you to not use generic collections in your case.

我对控件做了一些修改:

I've modified the Control a bit:

public class BTextBox : ItemsControl {

    static BTextBox() {
      DefaultStyleKeyProperty.OverrideMetadata(typeof(BTextBox), new FrameworkPropertyMetadata(typeof(BTextBox)));
    }

    private TextBox _textBox;
    private ItemsControl _itemsView;

    public static readonly DependencyProperty ProviderCommandProperty = DependencyProperty.Register("ProviderCommand", typeof(ICommand), typeof(BTextBox), new PropertyMetadata(null));
    public static readonly DependencyProperty AutoItemsSourceProperty = DependencyProperty.Register("AutoItemsSource", typeof(IEnumerable), typeof(BTextBox), new PropertyMetadata(null, OnItemsSourceChanged));

    public IEnumerable AutoItemsSource {
      get {
        return (IEnumerable)GetValue(AutoItemsSourceProperty);
      }
      set {
        SetValue(AutoItemsSourceProperty, value);
      }
    }

    private static void OnItemsSourceChanged(DependencyObject d, DependencyPropertyChangedEventArgs e) {
      var tb = d as BTextBox;
      if ((e.NewValue != null) && ((tb.ItemsSource as IList) != null)) {
        foreach (var item in e.NewValue as IEnumerable) {
          (tb.AutoItemsSource as IList).Add(item);
        }

      }
    }

    public override void OnApplyTemplate() {
      this._textBox = this.GetTemplateChild("PART_TextBox") as TextBox;
      this._itemsView = this.GetTemplateChild("PART_ListBox_Sugg") as ItemsControl;
      this._itemsView.ItemsSource = this.AutoItemsSource;
      this._textBox.TextChanged += (sender, args) => {
        this.ProviderCommand?.Execute(this._textBox.Text);
      };

      base.OnApplyTemplate();
    }

    public ICommand ProviderCommand {
      get {
        return (ICommand) this.GetValue(ProviderCommandProperty);
      }
      set {
        this.SetValue(ProviderCommandProperty, value);
      }
    }

    public ICommand AddCommand {
      get {
        return new RelayCommand(obj => {
          (this.ItemsSource as IList)?.Add(obj);
        });
      }
    }

  }

然后，我修复了您的XAML，以使它甚至可以编译和运行:

Then i've fixed your XAML to get thing to even compile and run:

<Style TargetType="{x:Type local:BTextBox}">
        <Setter Property="Template">
            <Setter.Value>
                <ControlTemplate TargetType="{x:Type local:BTextBox}">
                    <Border Background="{TemplateBinding Background}"
                            BorderBrush="{TemplateBinding BorderBrush}"
                            BorderThickness="{TemplateBinding BorderThickness}">
                        <Grid>
                            <Grid.RowDefinitions>
                                <RowDefinition Height="40"/>
                                <RowDefinition Height="Auto"/>
                            </Grid.RowDefinitions>
                            <TextBox x:Name="PART_TextBox" Grid.Row="0"  VerticalAlignment="Center" />
                            <ListBox ItemsSource="{TemplateBinding AutoItemsSource}" Grid.Row="1" x:Name="PART_ListBox_Sugg" HorizontalAlignment="Stretch" VerticalAlignment="Stretch">
                                <ListBox.ItemTemplate>
                                    <DataTemplate>
                                        <CheckBox Command="{Binding RelativeSource={RelativeSource FindAncestor, AncestorType={x:Type local:BTextBox}}, Path=AddCommand}" CommandParameter="{Binding}" Foreground="#404040">
                                            <CheckBox.Content>
                                                <StackPanel Orientation="Horizontal">
                                                    <TextBlock Text="{Binding }" Visibility="Visible"  TextWrapping="Wrap" MaxWidth="270"/>
                                                </StackPanel>
                                            </CheckBox.Content>
                                        </CheckBox>
                                    </DataTemplate>
                                </ListBox.ItemTemplate>
                            </ListBox>
                        </Grid>
                    </Border>
                </ControlTemplate>
            </Setter.Value>
        </Setter>
    </Style>

最后一个很有价值的评论:

At last a valuable remark:

永远不要在您的ItemsSources上使用二传手.如果覆盖它们，绑定将中断.如下所示，请使用.Clear()和.Add():

Never ever allow setters on your ItemsSources. If you override them, the binding will break. Use .Clear() and .Add() instead as you see below:

public class StringModel : INotifyPropertyChanged {
    public event PropertyChangedEventHandler PropertyChanged;

    private readonly ObservableCollection<string> _collection = new ObservableCollection<string>();
    private readonly ObservableCollection<string> _suggCollection = new ObservableCollection<string>();
    private readonly ObservableCollection<string> _primaryCollection = new ObservableCollection<string>();

    public ObservableCollection<string> Collection => this._collection;

    public ObservableCollection<string> SuggCollection => this._suggCollection;

    public StringModel() {
      this._primaryCollection.Add("John");
      this._primaryCollection.Add("Jack");
      this._primaryCollection.Add("James");
      this._primaryCollection.Add("Emma");
      this._primaryCollection.Add("Peter");
    }

    public ICommand AutoBTextCommand {
      get {
        return new RelayCommand(obj => {
          this.Search(obj as string);
        });
      }
    }

    private void Search(string str) {
      this.SuggCollection.Clear();
      foreach (var result in this._primaryCollection.Where(m => m.ToLowerInvariant().Contains(str.ToLowerInvariant())).Select(m => m)) {
        this.SuggCollection.Add(result);
      }

    }

  }

注意

因为我没有您的DelegateCommand实现，所以我改用了RelayCommand.您可以在遇到任何问题时进行更改.我认为它是一样的东西，但名称不同.
您也可以考虑从一开始就显示您的建议.这样可能会提供更好的用户体验，但这仅是我的意见

Sice i didnt have your DelegateCommand-implementation, i've used my RelayCommand instead. You can change it withour any issues. I think its the same thing but a different name for it.
You also might consider to display your suggestions right from the start. This might provide a better user-expierience, but thats just my opinion