No. The super facehugger in Alien 3 was laid by the queen while she was on the shuttle in Aliens.
The egg being created in Alien by the lone drone was a normal Facehugger. Had that drone succeeded, it would have then become queen after another xenomorph was present. The gender of that secondary drone does not truly matter: if it were a female, it could have created more eggs using the same process, if it were male then it would eventually mate with the queen.
And yes, I did make this up. The only references I had were what I have seen in the films and what H.R. Giger has illustrated. Nothing else.
The reason that unfertilized eggs bear queen facehuggers is that there is no Y chromosome to be added to the egg's genetic code. This creates a queen facehugger, which then becomes a queen after development. If the Y chromosome is present, then the egg develops into a standard facehugger.
"If anything in this life is certain, if history has taught us anything, it is that you can kill anyone." - Michael Corleone