In a 5- to 6-quart pan over high heat, bring 1 1/2 to 2 quarts water to a boil. Rinse and drain wild rice. Add to boiling water, cover, and return to a boil; reduce heat and simmer, covered, until rice is tender to bite and beginning to split, 35 to 45 minutes. Drain.
Meanwhile, cut off and discard pulp from squash. Cut squash into 1/2-inch cubes (you should have about 4 cups).
Remove casings from sausages and discard. Crumble sausages into a 5- to 6- quart nonstick pan; stir often over medium-high heat until browned, about 5 minutes. Discard all but 2 tablespoons fat in pan. Add onion, stir often until lightly browned, about 5 minutes. Add white rice; stir until beginning to turn opaque, about 3 minutes.
Add broth, currants, coriander, nutmeg, and cinnamon to pan. Bring to a boil over high heat, cover reduce heat, and simmer for 5 minutes. Add squash; cover and simmer over low heat, gently stirring once, until rice and squash are tender to bite, 15 to 20 minutes.
Squeeze liquid from spinach. Add spinach and wild rice to pan and mix gently. Cover and cook until hot, about 5 minutes. Add salt and pepper to taste.