FileMaker GO does not support the drop down calendar that you can use in FileMaker Pro. You get the rather awful "spinning wheels" date picker that's part of the iOS instead.
For that reason, I built my own Date Picker using a slide control inside a popover with some simple scripting. I set it up to be very easy to install into new files/layouts. Note that swiping the date picker horizontally changes the month displayed in the date picker.
I just realized that the date "spinning wheel" does show up on the 5s but not on the 5c...
Thanks, I will have a look at your work-around.
It should show up for both, thought I don't have a 5s to test. If you can get that result consistently with a brand new file and a field that is of type date, I suggest reporting it as a possible bug in Report an Issue.