I have a model bound to grid.MVC, every other filter works fine apart from the date picker filter.
I have added the grid.mvc.datepicker.css to the bundle and I am getting the date picker.

There are no errors in any of the js and all the js files load properly
And this is how I initiate the column in the view
columns.Add(o => o.VoucherType.Expiry)
.Titled("Expiry")
.SetFilterWidgetType("")
.Format("{0:dd/MM/yyyy}");
When I click on the date , nothing happens to the grid.
I was able to reproduce the problem you are experiencing.
Since Grid.MVC is based on bootstrap you need to include the following bootstrap scripts in your solution:
I reproduced your problem by removing the bootstrap-datepicker.js from my solution includes.
Thank you