I was eventually able to clear this problem by going to:

<State and Local (on the left)>

<Other>

<CA Like-Kind Exchanges (3840) - Annual/Final Data>

Scroll down to Part I "Date like-kind property given up was originally acquired"

Delete the date so that it is blank.

I then clicked preview forms, and it now shows 08/01/2007 instead of 16/02/4014.

I was then able to e-file CA successfully.  

I should also note that when I return to that screen now, it once again has 08/01/2007 in the filed, but the preview still shows correctly, so even though it now looks exactly the same as it did before in the entry field area, it is no longer bugged on the forms and filing.