Boil the summer squash in salted water until tender. Drain, pressing to squeeze out as much water as you can. Place in a bowl and mash. Add the Velveeta while the squash is still hot so that it will start to melt.
Melt the butter in a skillet or pan. Add the olive oil and the onion, red and green peppers along with the celery and saute until the celery is soft and the onion is translucent. Pour this mixture over your squash and Velveeta. Mix to combine. Add the poultry seasoning and salt and pepper to taste. Check the seasoning before you add the egg. When your seasoning is where you want it, add the beaten egg and start with about a half cup of the bread crumbs, adding more if needed.
Pour the squash mixture into a buttered casserole. Top with the cheese and cover with foil and cook about 20 minutes. Uncover and cook for an additional 5 minutes or so until the cheese is melted and starts to slightly brown.