I know how you feel. I killed a cobra in South Africa that was hooding at my Mum's little pomerianian. The dog was right in it's face barking like crazy and I had no choice, but man did I feel bad about killing one of God's creatures that was really only defending itself.
No worries on dispatching the squirrel....but I would be looking for the cause. Snake...poison....etc. Not normal for a tree rat to act in this manner.
Sometimes things don't work out as we intend. No reason for you to feel bad you handled the situation according to the circumstances presented. And like you said who knows what it could have had since it was acting so strangely I would have done the same thing.
I am sure that it was tough to do but sometimes we have to do things that make us feel bad. According to your description it was obviously sick or injured. Don't beat yourself up over it.
You "feel" like a bad person becuase our bleeding heart society has made you feel that if you take care of your land and family, and have to take out nature..you ARE a bad person..which is a very big load of bahooey. You did the right thing, period.
I am so sorry that you and your daughter had to go through this. It is horrible. But do not feel bad. You did what you to do what you had to to protect your daughter and you. Sending lot's of hugs