1. Put vinegar, salt, sugar, and 3 cups water in a saucepan and heat over medium-low heat until salt and sugar dissolve. Let cool.
2. Peel squash with a vegetable peeler. Cut in quarters crosswise and scrape out seeds. Cut quarters in half, then into 1/4-in.-wide slices. Cut in half again.
3. Pour enough oil into a large deep pot to come at least 3/4 in. up sides; heat to 325° on a deep-fry thermometer. Fry squash in batches, turning often, until tender and golden brown, 3 to 5 minutes. Drain on paper towels. Fry sage until crisp, about 45 seconds; drain. Fry garlic until golden, 1 to 2 minutes; drain. Save oil for another use.
4. Layer squash into a terrine or glass loaf pan (at least 1 1/2-qt. size). Top with garlic and sage and pour in enough cooled brine to cover by 1/2 in. Chill at least 1 hour.
Make ahead: Up to 1 day, chilled.