I had the same problem…and ended up deactivating the plugin and after a lot of research I found this site, followed it step by step (not too difficult at all) and it worked perfectly:
https://www.wordpressninja.com/2011/06/how-to-add-a-jquery-datepicker-to-your-contact-form-7/
The only thing I had to do was make sure the folder and one of the file names matched. Meaning when I downloaded the jQuery folder it was named “jquery-ui-1.10.2.custom”, so to make it easier I changed it to “jquery-ui-1” (to match the tutorial) and in the coding, the example refers to file name “jquery-1.5.1.js” which I had to change to the current version in my file folder. All in all very little editing.
Hope this helps anyone else having trouble.