Use the OfType method to filter objects based on their type, then use the Where method to filter on the name:
var query = objects.OfType<Employee>()
.Where(e => e.Name == "John")
.Select(e => e.EmployeeId);
This will return the employee IDs for all employees with the name of "John." If you expect only one person to match that criteria, you can replace the Where method with Single, or if you want to take the first result use First. However, if you expect one person but don't know whether they exist, you'll want to use SingleOrDefault:
Employee result = objects.OfType<Employee>()
.SingleOrDefault(e => e.Name == "John");
if (result != null)
{
Console.WriteLine(result.EmployeeId);
}