| The only location equipment grounds and neutrals are allowed to be connected together is in the enclosure that contains the Service Main Switch. If your panel has the Main in it, then it may not matter which goes where.
Even thou your panel may have a Main, the installer may have separated the two just because that is normal for us. Someone else coming along,if the Main is in that box, may be able to use both, really depends on the arrangement of the incoming Main neutral and Main system ground.You an only use the ground bar with neutrals if this bar is solidly connected to the Main neutral conductor.
A picture of the inside of the panel showing both the neutral and ground bar, I can interpret how it is arranged.
The reason is that the two are separated once they leave the Main, is to prevent any fault currents from flowing in the neutral, and develops a the
"Single Point Ground" system. Fault currents must flow thru a low impedance path with no other currents, and believe it or not, a neutral is a current carrying conductor.
Your right, working fine now is normal, the equipment ground is there for when circuits are not normal, such as a line to ground short. This is when , if the two are not separated properly, damage can occur.
When you say "same clamp" do you mean the screw terminal? Only if the terminal is rated to handle multiple conductors can more than one wire use this terminal. This can be confirmed by reading the specs of the panel, usually pasted on the door, and certainly found at the manufacturers website.
Since at this time I do not know how your panel is arranged, the GFI neutral pigtails must only be connected to a true neutral bar.
These details, and so much more, are the things lay people do not know, and can create scary situations by not doing electrical work correctly. I am glad you asked the question. Grounding is the most important aspect of electrical wiring. How many ground wires get cut off because they are believed to be useless. | 
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