BIT columns all "1" after a phpMyAdmin export/import
I think we'll need some more information to pin down your problem. I just tried to recreate your issue and I could not.

I created a table named bitTest with two columns:
textName VARCHAR(5)
bitValue BIT(1)  

I inserted two rows into the table using phpMyAdmin's "Insert" feature



Then I did a phpMyAdmin "Export" using the "Quick method". The resulting .sql file was

-- phpMyAdmin SQL Dump
-- version 3.4.10.1deb1
-- http://www.phpmyadmin.net
--
-- Host: localhost
-- Generation Time: Apr 01, 2013 at 03:34 PM
-- Server version: 5.5.29
-- PHP Version: 5.3.10-1ubuntu3.6

SET SQL_MODE="NO_AUTO_VALUE_ON_ZERO";
SET time_zone = "+00:00";


/*!40101 SET @OLD_CHARACTER_SET_CLIENT=@@CHARACTER_SET_CLIENT */;
/*!40101 SET @OLD_CHARACTER_SET_RESULTS=@@CHARACTER_SET_RESULTS */;
/*!40101 SET @OLD_COLLATION_CONNECTION=@@COLLATION_CONNECTION */;
/*!40101 SET NAMES utf8 */;

--
-- Database: `zzz`
--

-- --------------------------------------------------------

--
-- Table structure for table `bitTest`
--

CREATE TABLE IF NOT EXISTS `bitTest` (
  `textName` varchar(5) NOT NULL,
  `bitValue` bit(1) NOT NULL,
  PRIMARY KEY (`textName`)
) ENGINE=InnoDB DEFAULT CHARSET=latin1;

--
-- Dumping data for table `bitTest`
--

INSERT INTO `bitTest` (`textName`, `bitValue`) VALUES
('False', b'0'),
('True', b'1');

/*!40101 SET CHARACTER_SET_CLIENT=@OLD_CHARACTER_SET_CLIENT */;
/*!40101 SET CHARACTER_SET_RESULTS=@OLD_CHARACTER_SET_RESULTS */;
/*!40101 SET COLLATION_CONNECTION=@OLD_COLLATION_CONNECTION */;


Then I dropped the table and used phpMyAdmin's "Import" feature ro re-create and re-populate it using that .sql file. The BIT values were (correctly) imported as 0 and 1.

Perhaps you could update your question with some excerpts from the .sql file you exported to see what differences might exist between your export file and mine.