If you get the latest code I've exposed a few more properties on jDatePicker.cs to allow you to configure the showOn, constrainInput, and also it is now possible to configure an image for the button.
However, I kept the default behavior as it was. One problem with using showOn=both and/or constrainInput=true is that it does not allow you to leave the field blank like when you have a begin date and end date and you want to leave end date blank so there is no end date.
So we need to use it in a variety of situations in mojoPortal and the default settings are designed to make it meet those diverse needs. We are also somewhat constrained by our DatePicker wrapper control and provider model that allows plugging in different date pickers, this makes us have to use a common api exposing only the most common date picker properties on the control and it is difficult to expose all the various properties of a particular underlying datepicker. In your custom features you can of course use the jDatePicker.cs directly instead of using our wrapper control so you can have more control over properties available on the control.