First of all the MySQL "date" field type only accepts dates in the YYYY-MM-DD (that is (4digit year dash 2digit month dash 2digit day) format so you need :
$user_date = date('Y-m-d');
Watch your letter CaSe, it is important. Y in the PHP date() function is "Year as four digits" whereas y is "Year as two digits". M is "Abbreviated month name(Jan, Feb,...)" whereas m is "Month number from 01 to 12".
I think you get the picture!
Now just insert the var $user_date into the database just like you were before. The UNIX_TIMESTAMP() only comes into play when retrieving the date. PHP uses a unix timestamp for it's date() function. A shortcut to get MM/DD/YYYY would be.
SELECT DATE_FORMAT(datefieldname,"%m/%d/%Y") FROM dbname WHERE blah=blah ORDER BY datefieldname;
This way MySQL will convert the date from YYYY-MM-DD to MM/DD/YYYY, and you don't have to worry about the timestamp.
Hope this makes some sense,
Rodney