No.
Gil Travel does not assign airlines to your Birthright trip. The airline that your group is flying with is determined SOLELY by Taglit Birthright Israel and Gil Travel has no involvement in that part of your trip.
Birthright provides you with a full paid (or for some programs, a subsidized) ROUND TRIP ticket. That ticket is only on the airline that Birthright assigned your group to.
Regarding your return flight home, we, as the travel agency, can change the date (extension) – or in some cases the route – of your return ticket. But we cannot provide you a NEW ticket on a different airline. We only manage the airline ticket that Birthright assigned you to.
If you would like to travel on a different airline, you can forfeit your return ticket (see FAQ question on forfeiting your return), and purchase at your expense a new flight from any airline you choose.