Is it possible to change the colour of the line below / Border of a TextBox (Entry)

you can use custom renderer that will affect all entries,

here's for android:

[assembly: ExportRenderer(typeof(Entry), typeof(MyEntryRenderer))]
namespace Android.MyRenderers
{
    public class MyEntryRenderer : EntryRenderer
    {
        protected override void OnElementChanged(ElementChangedEventArgs<Entry> e)
        {
            base.OnElementChanged(e);

            if (Control == null || e.NewElement == null) return;

            if (Build.VERSION.SdkInt >= BuildVersionCodes.Lollipop)
                Control.BackgroundTintList = ColorStateList.ValueOf(Color.White);
            else
                Control.Background.SetColorFilter(Color.White, PorterDuff.Mode.SrcAtop);
         }    
    }
}

and iOS:

[assembly: ExportRenderer (typeof(Entry), typeof(MyEntryRenderer))]
namespace iOS.MyRenderers
{
    public class MyEntryRenderer : EntryRenderer
    {
        private CALayer _line;

        protected override void OnElementChanged(ElementChangedEventArgs<Entry> e)
        {
            base.OnElementChanged (e);
            _line = null;

            if (Control == null || e.NewElement == null)
                return;

            Control.BorderStyle = UITextBorderStyle.None;

            _line = new CALayer {
                BorderColor = UIColor.FromRGB(174, 174, 174).CGColor,
                BackgroundColor = UIColor.FromRGB(174, 174, 174).CGColor,
                Frame = new CGRect (0, Frame.Height / 2, Frame.Width * 2, 1f)
            };

            Control.Layer.AddSublayer (_line);
        }
    }
}

not sure about Windows solution on this

I had the same problem and just changing the colorAccent value in styles.xml (in Xamarin.Android project) will change the colour of the cursor and the bottom border of an Entry field.

<item name="colorAccent">#BA55D3</item>

Since I have the content pages with one background color and dialogs with another, using Styles to specify the bottom bar color is totally the wrong answer. And since the OP only asked about Android, this is just Android...

I use a custom renderer to set the bottom bar color the same as the text color. Have to both ElementChanged and PropertyChanged.

[assembly: ExportRenderer(typeof(Xamarin.Forms.Entry), typeof(CustomEntryRenderer))]
namespace XamFormsConnect.Droid
{
    public class CustomEntryRenderer : EntryRenderer
    {
        protected override void OnElementChanged(ElementChangedEventArgs<Xamarin.Forms.Entry> e)
        {
            base.OnElementChanged(e);

            if (Control != null && e.NewElement != null)
            {
                var entry = (Xamarin.Forms.Entry)e.NewElement;
                if (Build.VERSION.SdkInt >= BuildVersionCodes.Lollipop)
                    Control.BackgroundTintList = ColorStateList.ValueOf(entry.TextColor.ToAndroid());
                else
                    Control.Background.SetColorFilter(entry.TextColor.ToAndroid(), PorterDuff.Mode.SrcAtop);
            }
        }

        protected override void OnElementPropertyChanged(object sender, PropertyChangedEventArgs e)
        {
            base.OnElementPropertyChanged(sender, e);

            if (e.PropertyName == "TextColor")
            {
                var entry = (Xamarin.Forms.Entry)sender;
                if (Build.VERSION.SdkInt >= BuildVersionCodes.Lollipop)
                    Control.BackgroundTintList = ColorStateList.ValueOf(entry.TextColor.ToAndroid());
                else
                    Control.Background.SetColorFilter(entry.TextColor.ToAndroid(), PorterDuff.Mode.SrcAtop);
            }
        }
    }
}