It's not so much that the oils are bad persay but more like they're empty calories. Much like for humans, if you want the body to function well you want to avoid as many empty calories as possible.
A chicken eating a lot of empty calories may gain weight and lose condition at the same time. An overweight hen is much more prone to reproductive and egg laying issues.
Ergo; oils = bad in excess and it's better to just avoid them because the calories they add are meaningless and they get all the calories they need from the feed they eat.
On their own or in moderation oils are pretty harmless. In fact they can be used without harm to treat impacted crop. My dog once drank a full quart of cooking oil by accident and was fine aside from some diarrhea and vomiting (IE, a VERY upset tummy). So some oil is fine, it's just better to avoid it.
Also, remember that articles on this site are written by regular people and usually aren't entirely comprehensive and just one perspective/one persons knowledge. Same with posts.
As for variety... Generally speaking squash is squash. Especially winter squash is winter squash and summer squash is summer squash. And unless there's something specific about the squash that makes is no good (like I probably wouldn't feed a luffah sqaush) , pretty much all squash varieties are fine.
