Put the grated zucchini onto a clean tea towel and squeeze out as much liquid as possible. Measure out 2 cups of the squeezed zucchini and set it aside.
Preparing the batter
In a medium bowl, whisk together gluten-free flour, salt, baking soda, baking powder, and ground cinnamon. Set it aside.
In a large bowl, add the eggs, granulated sugar, brown sugar, oil, Greek yogurt, vanilla extract, and mix together until smooth.
Add the flour mixture and squeezed zucchini to the large bowl of wet ingredients. Mix until combined, and run a silicone spatula to make sure the sides and bottom of the bowl are free of pockets of dry flour. Let the batter rest for 15 to 20 minutes.
Baking
While the batter rests, preheat the oven to 350°F (175°C). Grease and line a 8.5 inch loaf pan with parchment paper.
Pour the batter into the prepared pan, and bake for 62 to 65 minutes, or until a toothpick inserted into the center comes out completely clean, or with a couple of crumbs on it. Let the zucchini bread sit in the pan for 15 minutes before taking it out onto a wire rack to cool completely.