As noted above, one shot of grease (or two or three for that matter) will not do any good. Looking at a cut away view of the Dexter EZ Lube axle (see link below) you can see that the grease goes in thru the outer port, thru a channel to the inner bearing, thru that bearing into the center of the hub, fills that void, and then exit out thru the outer bearing. Takes a lot of grease to do that. Unless you see grease exiting thru the outer bearing around the tip of the grease gun, you have not greased the outer bearing at all.
There are two schools of thought regarding the EZ Lube axles. One is do not use the grease zerk at all. Only hand pack.
Second is to very carefully follow the Dexter lubing instructions. The instructions say to very slowly pump grease into the zerk while constantly turning the wheel until you see grease come out the front. Turning the wheel allows grease to more easily flow thru the inner bearing and less pressure on the seal. Continue that until you see new grease come out the front indicating all the old grease has been replaced by new grease. Done carefully the grease does not go thru the rear seal. I have done that with no problem. Others have reported grease getting thru the seal on the brakes.
Hand packing is the best idea since the bearings need regular inspection anyway.
EZ Lube Axle