Mysql won't be able to handle your html form post. You will need to use a server-side language to check for those checkboxes being set and then modify your query with additional conditions.
You would likely get a better response if you were more specific, but I'll throw you an example with php. Assuming your form looks something like:
<form method="post">
<input type="checkbox" name="field1_cb" value="1" />
<input type="text" name="field1_txt" />
<br /><br />
<input type="checkbox" name="field2_cb" value="1" />
<input type="text" name="field2_txt" />
</form>
Your server code might look something like:
$query = "SELECT * FROM tbl WHERE field0 = 'asdf' ";
$fields = array('field1','field2');
foreach($fields as $f){
if(isset($_POST[$f.'_cb'])){
$query .= " AND $f = ".$_POST[$f.'_txt'];
}
}
//Run your query...
Please don't copy and paste this code directly, it is disgustingly insecure. Just trying to get you started.