Does my creature still get a +1/+1 counter?
No, your creature does not get a counter.

The exact meaning of "regenerate" is this (from rule 701.13a):


  If the effect of a resolving spell or ability regenerates a permanent, it creates a replacement effect that protects the permanent the next time it would be destroyed this turn. In this case, “Regenerate [permanent]” means “The next time [permanent] would be destroyed this turn, instead remove all damage marked on it and tap it. If it’s an attacking or blocking creature, remove it from combat.”


So, in your situation, the 2/2 would be destroyed, but instead, you remove all damage attached to it, tap it, and remove it from combat. This means that it doesn't die, so your creature doesn't get a counter.