I followed instructions I found in another post on creating a new page and placing the registration module on it to make a secure registration page.  This works fine.  However, I modified the Site Settings / User Account Settings to use a Registration Form Type of Custom and I specified Registration Fields.  

If I change the Registration Page back to <None Specified>, I can see the registration fields I specified, but this page can't be secured.  If I chaneg it back to use my custom page with the registration module, it is secured, but doesn't show the custom fields.  I have tried this on two different sites running DotNetNuke 6.2.6 and I get the same result.

Is there any way to secure the default Registration page or to use a custom page and display custom fields?

Have you tried upgrading to DNN 7.x? I'm using DNN 07.01.00 and the Registration Fields are retained in the Custom Registration Page.
Hope this is not too late yet...