For the pickles: Put the squash in a medium bowl. Combine the vinegar, sugar, pickling spice, dill, 3/4 teaspoon salt and 1/2 cup water in a small saucepan and bring to a boil. Pour over the squash and let stand 30 minutes at room temperature.
For the sandwiches: Meanwhile, lightly toast the bread, then cut out rounds with a 2-inch cookie cutter (discard the trimmings). Use a slightly larger cutter (about 2 1/4-inches) to cut out rounds from the bologna.
Melt the butter in a large nonstick skillet over medium-high heat. In batches, add the bologna and cook, turning once, until golden brown, about 1 minute per side. Transfer to a cutting board and cut in half.
Spread the mustard onto the bread rounds. Top each round with a folded half-slice of bologna, followed by a few slices of pickled squash and sprig of dill.