You can use DayPilotNavigator.BeforeCellRender event handler to add a custom CSS class to a day cell.
http://doc.daypilot.org/navigator/cell-customization/
Note that in version 7.7 the CssClass value will be prefixed with Theme/CssClassPrefix in the CssOnly. This is not consistent with the BeforeCellRender behavior in other controls and it will be fixed in the next release.
Example:
protected void DayPilotNavigator1_BeforeCellRender(object sender, DayPilot.Web.Ui.Events.Navigator.BeforeCellRenderEventArgs e)
{
if (e.Start.DayOfWeek == DayOfWeek.Friday)
{
e.CssClass = "weekend";
}
}