I'm sorry no one has answered you yet. I don't have a lot of experience with it, despite having a large family, because we were not prone to allergies and I delay the introduction of solids and then do more of the "a little from mom's plate" approach rather than single baby foods.
How old is your daughter?
Your description of what she did with oatmeal does sound like my oldest son, when we were starting him on some solids. He was a "gurker" - a very very spitty uppy kid, the kind that when I took him to the church nursery they'd all back away, not wanting the one who would gurk on them LOL - but also a happy spitter.
Anyway, when he was 8-10 months old and starting oatmeal it would make him throw up. I do not believe it was an allergic reaction, but I do think that his stomach/esophagus was just not ready for it. He struggled with any larger amount of solid food, but particularly thicker things like oatmeal and rice cereal, until he was about a year. He was not diagnosed with GERD, but I am wondering in retrospect if he might have had some issues - I don't think he was in pain or had damage, but I think maybe he might've had some reflux / an immature esophagus and just didn't hold things *down* well until he hit a year.
As far as your specific question, I think *typically* the answer is yes, if they're fine with one winter squash the rest should be okay. But I know there are some people with allergies who are more sensitive than others. So I would perhaps keep a log, just in case, and I would not carefully wait when introducing the 2nd, 3rd, and 4th type squashes, but maybe just keep that info tucked away in case she reacts, then you don't have to backtrack much ...
I hope that helps a little.
