Your checkboxes should probably have quotes around your attributes and should have some value other than 'checked.' I usually use a value of 1:
echo "<input type=\"checkbox\" name\"=$id\" value=\"1\">$value";
Generally speaking, when you want to set a checkbox as checked, you set the attribute checked equal to "checked":
$check_the_checkbox = TRUE; // set this to false and it won't be checked
echo "<input type=\"checkbox\" name\"=$id\" value=\"1\"" . ($check_the_checkbox ? ' checked="checked"' : '') . ">$value";
that whole bit in parentheses with the ? and the : is shorthand notation for an if/then statement.
you could just as easily check a value in $REQUEST instead:
echo "<input type=\"checkbox\" name\"=$id\" value=\"1\"" . (($_REQUEST[$id] == 1) ? ' checked="checked"' : '') . ">$value";