I think you'd have to be very lucky for that to work. The seat is more or less a piece of tubing pressed into the carburetor body. The B&S-recommended process tries not to expand the (fairly thin-walled) seat, and extracts it by running a screw-thread through its central hole, then having the screw extend past the end of the seat so it can push against the carburetor body at the end of the blind hole surrounding the seat. This applies a purely axial force to extract the seat. An ezy-out is intended to remove screwed-in components, by unscrewing them. In the process it expands them considerably, which expands the hole they are in also. This may not matter with screwed-in components, but expanding a pressed-in component just makes it harder to get it out - and the ezy-out does not apply an axial force anyway, it just applies a rotational force.
I'd do this job by finding the size of the through-hole in the seat, and grabbing a fine-thread tap that has a root-diameter slightly smaller than that. I'd cut a thread on the inside of the seat with that tap, then run a screw with the same thread into the tapped seat and out the far side, thus using the B&S extraction process without needing their weird American screw. I hope whoever Nathan gets to help him will do something similar, if he doesn't have B&S screws readily to hand. Actually I'd do it my way even if I had the B&S screw: never follow the instructions if it's more fun to make something up.
