1. Wash vegetables. Score cucumber and squash lengthwise with a fork, leaving furrows in the peel on all sides. (This makes scalloped edges when vegetables are sliced.) Trim stem and blossom ends of cucumber and squash; cut into 1/8-inch slices. Place in a colander in sink; sprinkle with 2 Tbsp. salt, and toss gently. Let drain 30 minutes.
2. Meanwhile, peel carrot, and cut carrot and radishes into 1/8-inch-thick slices. Toss together with drained cucumber and squash.
3. Place 2 dill sprigs in each of 2 clean (1-pt.) jars or nonreactive containers with lids. Pack vegetables in jars, leaving 1/2-inch headspace.
4. Bring vinegar, next 3 ingredients, remaining 2 Tbsp. salt, and 2 cups water to a boil in a 1 1/2-qt. stainless steel saucepan over medium-high heat, stirring until sugar and salt dissolve. Pour hot vinegar mixture over vegetables to cover. Apply lids. Chill 24 hours before serving. Store in refrigerator up to 3 weeks.
Total time: 1 hour, plus 1 day standing time