(Grin.) Now that you mention it, I’ll post my reply here so everyone else can read it, too!
I recommend you taste the unripe and ripe fruits of any squash you’re thinking of saving seeds from, so you know whether you like it best as a summer squash, winter squash, or both. Both can absolutely happen, and squashes that are suitable both ways are my favorites. (Grin.)
You’re welcome to pull out anything you don’t like. You’re welcome to put in anything you like. If you add in a bunch of genetic diversity that wasn’t there before, great! That’s a bonus!
As for what to look for, it’s very simple: Whatever you like! If you like a plant for some reason, somebody else probably will, too. Just go ahead and save seeds from whatever plants you like, and you’ll be doing it perfectly. 
So, in answer to your other questions, which are great questions to ask.
In most plants, including squashes (and other cucurbits, like cucumbers, melons, and watermelons), the fruit and the shape of the seeds is all maternal tissue. So you’re correct, you’re only evaluating the mother based on the fruit and appearance of the seeds. You won’t see effects of crossing until you plant those seeds and can evaluate the child as its own plant.
There are a few species that exhibit crossing in the fruits or seeds themselves, but it’s rare. It’s called the xenia effect. In case you’re curious, I only know of two: corn and passionfruit. The color of corn kernels will indicate whether they’ve been crossed, and the fruits of passionfruits can sometimes change size based on who the father was. Very weird and surprising.
Otherwise, you can probably assume you’re only evaluating the mother when you look at seeds and fruits. This is true of legumes (beans / peas / etc.), cucurbits (squashes / melons / etc.), and most other plant families.
Yeah, that’s how I evaluate summer squash, too! I’m trying to leave the first two fruits on any plant to grow to maturity, and then I eat all the fruits afterwards as summer squashes. There are two benefits to doing this, both of which are likely to be universal:
Benefit #1: When you save seeds from the earliest fruits, the next generation will learn that setting fruits early is more likely to result in successful reproduction. When you save seeds from later fruits, the next generation will learn that waiting to set fruits until later is more likely to result in successful reproduction. Since early flowering and fruit set is usually highly desirable, you probably want to save seeds from the earliest fruit.
(Side bar: The reverse is true if you want late flowering. This is often true with leafy greens that taste better before they bolt, such as lettuce. With those, it’s often a good idea to pull out and eat all the plants that are trying to flower early, and then save seeds only from the slowest plants to bolt.)
Benefit #2: When you leave the earliest fruits (or seed pods) on the plant and only eat the later ones, you are selecting for plants that are happy to produce more food even after they’ve successfully ripened mature seeds. In most cases, especially in the case of things that are usually eaten unripe (like cucumbers, snap peas, and summer squashes), this is a highly desirable trait.
If you want to enhance the effects of that second benefit, you may want to save seeds only from plants that produce the most fruits (or seed pods) that are set after the first one or two (or ten or twenty, with something like beans), which are left on the plant to continue to mature. But if that sounds unappealing or too much of a bother, there’s no reason you have to.
Generally with seed saving, you get more of whatever you incentivize, so it’s often smart to consider whatever the plant’s point of view is. Fill in the blank in the following question:
“My descendants are more likely to survive if . . .”
There could be a whole lot of different answers to that question, so feel free to answer everything you think may be accurate. Here are a whole lot of ideas, to get the creative juices flowing.
- I can survive very little water, even if it means lower yields.
- I assume I’ll get lots of water and grow very rapidly and produce enormous yields.
- I am good at forming relationships with beneficial microbes in the soil.
- I prioritize growing big roots before making the leaves big.
- I prioritize growing big leaves before making the roots big.
- I ripen fruits quickly.
- I set fruits early.
- I grow fewer thorns.
- I grow bigger thorns.
- I taste bad.
- I taste good.
- A common disease in this climate doesn’t hurt me.
- A common herbivorous insect in this climate doesn’t want to eat me.
- My seeds germinate really easily.
- My seeds have inconsistent germination with complicated dormancy mechanisms. (This is usually better for survival for wild plants, because their seeds will be scattered among all kinds of unpredictable growing conditions.)
- My seeds only germinate after going through winter, so they’ll know when it’s spring and safe to come out.
- I can survive hotter temperatures.
- I can survive colder temperatures.
- I prioritize lots of flowers / leaves / roots / seeds / stems / etc.
- I prioritize larger flowers / leaves / roots / seeds / stems / etc.
Things will be much easier for you in the future if you teach your plants to want the same things you want. And there’s no wrong answer about what you want! 
Here are some fun examples of humans who want different things form their plants:
In my climate, I tend to prioritize plants that are drought tolerant (even though that tends to mean lower yields), because water is scarce in my summers. If water is abundant in your summers, and you notice you get higher yields by making sure the plants are watered on a regular schedule, you may want to prioritize the opposite.
Some people only want blackberries with a 10/10 for flavor, and they don’t mind thorns. I don’t want thorns, and I’m content blackberries with an 8/10 for flavor. They would save different seeds than I would. That’s great!
Some people want Brassica oleracea that flowers very quickly and makes a lot of flowers. (We call that broccoli or cauliflower.) Some people want Brassica oleracea that takes a very long time to flower and produces lots of leaves. (We call that cabbage, brussels sprouts, kale, or collard greens.) Some people want Brassica oleracea that puts all its effect into growing big fat stems and puts very little energy into flowers or leaves. (We call that kohlrabi.) Some people want Brassica oleracea that stays alive for many years and either flowers every year or doesn’t flower at all. (We call that tree collards or perennial kale.) Some people want Brassica oleracea to be lovely to look at, and don’t care what the flavor is. (We call that ornamental kale.)
Those are all the same species, and humans have produced a whole lot of “different” vegetables from the same wonderful species just by different people deciding what they like best and saving seeds in their own way. Isn’t that cool?
That’s why it’s so much fun to save seeds for yourself – you’re in charge of deciding what plants you want more of.
Ha ha, I hope my longwinded explanation helps. 
