In saucepan of boiling water, cook quinoa according to package instructions. Stir in pine nuts, sage and half each of the salt and pepper.
Meanwhile, in nonstick skillet, heat 1 tbsp of the coconut oil over medium-high heat; cook mushrooms, onion, garlic and remaining salt and pepper, stirring occasionally, until softened and no liquid remains, about 12 minutes. Add mustard and vinegar; cook, stirring, for 1 minute.
In small microwaveable bowl, microwave remaining coconut oil until melted.
Place 1 sheet of the phyllo on work surface with long end facing you; keep remainder covered with damp towel (to prevent drying out). Lightly brush phyllo with some of the remaining coconut oil. Top with second sheet of phyllo; lightly brush with some of the remaining coconut oil. Repeat layers with remaining phyllo and some of the remaining coconut oil.
Spoon quinoa mixture into 5-inch wide strip over top of phyllo, leaving 2-inch border at each short end and 2-inch border at long end farthest from you. Top with mushroom mixture, red peppers, strained tomatoes and arugula. Fold in short ends over borders. Starting from long end closest to you, roll up tightly.
Place strudel, seam side down, on parchment paper–lined rimmed baking sheet. Brush phyllo with remaining coconut oil. Bake in 400°F oven until golden, 25 to 30 minutes. Let cool on pan for 5 minutes. Slice crosswise into 6 pieces.
Tip from The Test Kitchen: Red quinoa boasts the same nutritional value and nutty, earthy flavour as white quinoa while also adding a vibrant, festive colour to dishes.