Try an impact driver first. You beat the end with a hammer and it converts the beating motion to a rotary motion and gets the screw out.
Harbor Freight sells one.
Also, if you know someone with an ultrasonic cleaner, it might be able to vibrate the jet enough to where it comes out easier.
If you drill, Harbor Freight sells a reverse drill bit set. But drilling should be a last resort. Its usually what one does right before he orders a new carb from ebay.
Another thing. Whoever reassembled the carb should not have used silicone sealer on the gaskets. Silicone sealer is not for use with gasoline. The gaskets work fine on their own. If you need help getting them to stick in place while you assemble it, use a thick grease.
Whatever was left of the sealer may be responsible for the current damage. If it is, then using a long tipped pencil type soldering iron to heat the jet may cause it to release the bond.