I am wondering if there exists a kind of epoxy or resin that doesn't dry "hard". I've seen so many of the hard variety in stores and online. The thing is, somewhere, somehow, I've gotten the idea that there is a kind that is sort of flexible, or squishy? Does such a thing exist? I really appreciate your help with this!
I'd say the short answer is no. "Epoxy" is the name of a 2-component artificial resin that always cures hard (at least when it's mixed and cured properly).
You can make squishy toys at home, but as far as I know there is no material involved that would be called a "resin" or even "epoxy". There are DIY kits available online that contain all the materials you need, or you can follow various online tutorials and get the ingredients yourself. Here's just a random tutorial by MEL Science (not affiliated) that shows 3 different recipes based on candle gel, alginate and silicone sealant. Each one has a slightly different consistency and some tear more easily than others. By searching for DIY squishies you can find countless other tutorials.
Another material that comes to my mind is molding silicone. It feels similar to epoxy in that it's also a 2-component material, but chemically it's completely different. It stays elastic after it's cured and well suitable to making squishy toys.