You can use BeforeCellRender event to highlight the current day. See also cell customization.

ASP.NET WebForms

protected void DayPilotScheduler1_BeforeCellRender(object sender, DayPilot.Web.Ui.Events.BeforeCellRenderEventArgs e)
{
  if (e.Start.DayOfWeek == DaysOfWeek.Saturday || e.Start.DayOfWeek == DaysOfWeek.Sunday)
  {
    e.BackgroundColor = "silver";
  }
}

ASP.NET MVC

Use OnBeforeCellRender method:

protected override void OnBeforeCellRender(BeforeCellRenderArgs e)
{
  if (e.Start.DayOfWeek == DaysOfWeek.Saturday || e.Start.DayOfWeek == DaysOfWeek.Sunday)
  {
    e.BackgroundColor = "silver";
  }
}

JavaScript

dp.onBeforeCellRender = function(args) {
  if (args.cell.start.getDayOfWeek() === 0 || args.cell.start.getDayOfWeek() === 6) {
      args.cell.backColor = "silver";
  }
};