I am at a topology conference today, and among the many good topologists here is Jan van Mill, a leading expert on topological groups. I ran your question by him, thinking he might know the answer off the top of his head. He did -- the answer is that if $G$ is a paracompact topological group, then $G \times G$ need not be paracompact.

The construction of such a group can be found in

Yinhe Peng and Liuzhen Wu, "A Lindelöf group with non-normal square" (link)

Their main theorem is that there is a Hausdorff group $G$ such that $G$ is Lindelöf but $G \times G$ is not. Every Hausdorff group is regular, every regular Lindelöf space is paracompact, and every Hausdorff paracompact space is normal. Thus their group $G$ is paracompact, while $G \times G$ is not.

More information on Peng and Wu's work can be found in these slides of Yinhe Peng's.