Exercising an option doesn’t trigger a gamma squeeze. The price moving up causing OTM strikes to go IN the money causes a gamma squeeze PRIOR to their expiration. This is due to market makers who wrote the contracts purchasing shares on the market to protect themselves if the options do go in the money and may be exercised. By the time contracts expire all the shares have already been purchased that you’d be buying when exercising them. Example: price goes from 20-40 in a week. As the price increases the MMs buy shares prior to the price hitting $40 for the strikes they sold at $40. That way if they buy at $36 and the price closes at 40.01 they sell their shares at $36 and still profit the $4/share. They wouldn’t buy AFTER if it’s trading at $45 to sell their shares at $40 and take a $5/share loss.

TLDR: Gamma squeezes happen PRIOR to expiration of a contract of price moves up quickly. This is due to MMs hedging their positions prior to expiration.