Preheat oven to 400°. Place 1 sheet of phyllo dough on a work surface; brush with butter. Layer with 4 additional phyllo sheets, brushing each layer. (Keep remaining phyllo covered with a damp towel to keep it from drying out.) Cut layered sheets crosswise in half, then lengthwise in half. Cover stacks to keep them from drying; repeat with remaining phyllo and butter.
Place each stack in a greased muffin cup. Fill each with a piece of cream cheese. Carefully break 1 egg into each cup. Sprinkle with salt and pepper; top with ham, cheese, bread crumbs and chives. Bring phyllo together above filling; pinch to seal and form bundles.
Bake until golden brown, 15-18 minutes. Serve warm.