What is the best method for saving spaghetti squash seeds?
Be sure the squash is completely ripe. When you cut open the squash before cooking, scoop out the seeds and any attached membrane [around the seed]. DH rinses the membranes off, I don't. Spread the seeds flat onto several layers of newspaper with another couple pages over them to keep them in the dark. Check twice daily and remove any damp newspaper. Depending on ambiant humidity, they should be dry within a few days. If you left the membranes, they will also be dry and can be discarded. I store my seeds wrapped in a dry paper towel, put in a labelled jar [close lid tightly], and then in the bottom of the refrigerator.
Next spring I will plant the plumpest seeds. If you are dubious about the seed quality, you can pre-sprout and then plant the best sprouts.
Meldy's instructions sound right on and are the way that I save my heirloom winter squash seeds, but I don't save my veggie spaghetti---'Orangetti', or this year, 'Hi-Beta Orange' seeds because they are hybrids. if you are growing the regular spaghetti squash, follow Meldy's instructions and you should be successful. However, if you are growing any hybrid squash, don't bother saving the seeds as they will not come true to type.
Also note that "sphagetti" squash are C. pepo and they cross readily with other members of the species like Crooknecks, patty pans, zucchini and pumpkins. Hand pollination and bagging of blossoms may be in order.