Crostini with roasted squash puree, carmalized onions, and goat cheese. The pioner woman has a good pureed squash recipe.

How about some tasty little fritters?

Beet chips with some tasty dip?

I did these sweet potato and sausage skewers with lemon garlic mayo once. The mayo was so good! We still make it sometimes just to have on hand.

http://www.myrecipes.com/recipe/sweet-potato-squares-with-lemon-garlic-mayonnaise-10000001831906/

Humble Bean's smashed kabocha on toast, with caramelized onion and ricotta
http://www.humblebeanblog.com/2012/11/smashed-kabocha-on-toast/

You could cube and then roast them, and then serve with a creamy dip

And toothpicks, of course

Try these Root Veggie Kettle Chips!
http://theviewfromthegreatisland.blogspot.com/2012/11/minimal-monday-root-veggie-kettle-chips.html

Does a dip count as a finger food? I made the butternut squash-parmesan dip from this site, served with pita chips, and it's been a huge hit every time I've made it!

Duchess Potatoes are pretty and can be bite sized and on a tray. Grilled squash and carrots on mini skewers drizzled with olive oil and rosemary. Bake those beets into a chocolate muffins (i had a recipe but can't find it, sorry).

You could make a roasted veggie soup and serve it in shooters, or cube up roasted beets and make skewers with the roasted beets, chunks of feta or ricotta salata and some fresh herbs. You could also make chips with the turnips and beets. Here is the link to the original recipe from Jean-Georges Vongerichten's kabocha squash on toast which was featured in the times which was absolutely DELICIOUS. I made it on Thanksgiving. http://www.nytimes.com/recipes/12944/Jean-Georges-Vongerichtenrsquos-Squash-on-Toast-.html

prosciutto-wrapped roasted butternut squash.
Parsnip/potato pancakes with barbeque brisket and horseradish cream.

How about "non-sausage" rolls (they're vegetarian) with butternut squash and goat cheese.
http://outoftheordinaryfood.com/2011/11/11/non-sausage-rolls-with-butternut-squash-and-goat-cheese/

Or parsnip and ginger pakoras with a dipping sauce...
http://outoftheordinaryfood.com/2012/01/27/parsnip-and-ginger-pakoras/

Or little, bite-sized carrot and apple dumplings in a rosemary crust...
http://outoftheordinaryfood.com/2011/11/25/roasted-carrot-apple-with-goat-cheese-in-a-rosemary-crust/

These are delicious! And not hard to make, with won ton wrappers from the grocery store. Crispy roasted beet and caper wontons with a port wine/sage sauce
http://outoftheordinaryfood.com/2012/04/22/crispy-beet-caper-wontons-with-port-wine-sage-sauce/

Love love loving all these ideas! Thanks!