Question

1.How do I use PrepareStatement for familyname and givenname?
2.Also, how do I case insensitive search by familyname or givenname?

String query ="SELECT agent.familyname, agent.givenname" +
                    " FROM agent" +
                    " WHERE agent.agentid = piececreation.agentid" +
                    " AND (LOWER(familyname) = '"+agent_lastname+"' OR LOWER(givenname) = '"+agent_name+"') ORDER by familyname";


            PreparedStatement pst = conn.prepareStatement(query, Statement.RETURN_GENERATED_KEYS);
            pst.setString(1, agent_lastname);
            pst.setString(2, agent_name);
            // Executing the insert
            pst.executeUpdate();

Answer 1

Make familyName or givenName to lowercase too since you are already using LOWER DB API

String query ="SELECT agent.familyname, agent.givenname" +
                        " FROM agent" +
                        " WHERE agent.agentid = piececreation.agentid" +
                        " AND (LOWER(familyname) = '"+agent_lastname.toLowerCase()+"' OR LOWER(givenname) = '"+agent_name.toLowerCase()+"') ORDER by familyname";

When you are using PreparedStatement dont append values directly in your SQL, if you do that you are prone to SQL Attack instead parametrize your values.

String query = 
                "SELECT agent.familyname, agent.givenname"
                + " FROM agent"
                + " WHERE agent.agentid = ?"
                + " AND (" 
                + " LOWER(familyname) = ? OR LOWER(givenname) = ?" 
                + ") " 
                + " ORDER by familyname";


        pst.setInt(1, piececreation.agentid);
        pst.setString(2, agent_lastname.toLowerCase());
        pst.setString(3, agent_name.toLowerCase());

Then set values calling appropriate setXXX methods as defined here.

You can read tutorial here