Can you tell me what kind of tool it takes to tighten up the squeeze type battery terminals?

Hello,

There is not any special tool specific for these type terminals.
If the pinching/squeezing force is too weak and the connector stays loose then best option is to replace the terminal. These are designed to squeeze the terminal as the nuts gets tightened.

Is the connector moves or loose on the battery terminal?

Let me know what you got.
Thank you.