A gathered frock for kids with a square neckline and sleeveless design, made of cotton satin cloth in white and green, would be a lovely choice! Here's a description of such a frock:

Color Scheme: The frock would feature a combination of crisp white and a vibrant shade of green. The white could be the primary color of the frock, while green accents or detailing would add a refreshing pop of color.

Fabric: Cotton satin cloth offers a luxurious feel and a subtle sheen, making it perfect for special occasions. It's lightweight, breathable, and comfortable for kids to wear, ensuring they can move freely and comfortably throughout the day.

Square Neckline: The frock would boast a sophisticated square neckline, which adds a touch of modern elegance. This neckline frames the shoulders beautifully and provides a flattering silhouette for children of all ages.

Gathered Design: Incorporate a gathered waistline to add fullness and create a charming silhouette. The gathered waist can be accentuated with a coordinating fabric belt or bow in the green shade, adding visual interest and defining the waist.

Sleeveless Design: A sleeveless design is perfect for warmer weather or indoor events, allowing kids to stay cool and comfortable while showcasing their arms. It also adds a playful and youthful touch to the frock's overall aesthetic.

Details: Consider adding delicate embellishments such as floral appliqués, lace trims, or fabric buttons in the green color to enhance the beauty of the frock. These details can be strategically placed along the neckline, waistline, or hem for added charm.

Overall, a gathered frock with a square neckline and sleeveless design in white and green would be a stylish and charming choice for any special occasion or casual outing. The combination of luxurious fabric, modern neckline, and vibrant color palette creates a timeless ensemble that any child would adore wearing.