I think of a stop and go as move when you are OOP against a bet/raise. Instead of reraising (usually a push-this is kind of a stack depend move), you call with the intention of pushing on the next street no matter the card that falls since you will be first to act. You may have more fold equity with a stop and go vs a reraise, especially if the card that comes is a scare card.
The situation you describe seems more like not wanting to get into a potential coinflip with JJ than a stop and go.
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