First, a disclaimer. I have never transferred from a flight arriving at EZE from the USA to Santiago. I have transferred to a flight to Uruguay. Passengers were ushered to a waiting room that did not require us to clear customs or to pay the exit tax. I assume the same is true for transfers to other countries, but can't confirm it....
You will not have to leave the int'l area and, accordingly, you will not have to pay the departure tax.
The airport EZE is very modern and professional. Assuming nobody is on strike you will have no problems. There will be sufficient English speakers about to help you with any questions or transfer questions. Ironic you are going from BA to Santiago. I generally am going the opposite way. I fly LAN Chile to Santiago and then connect there to BA. Several times I was able to get LAN Chile flights from NYC to BA at a low cost. Much cheaper than Aerolineas Argentinas. Now I travel American Arlines which has been better on price than all the national airlines other than some of the small ones like Avianca. The Santiago Airport is a nice little airport by the way. The flight over the Andes is spectacular so get a window seat if you possibly can. Have fun.