It's not what they're made of so much as how they're made. It's probably still Alder or Ash, but it'll be a glue-up of some number of pieces or pieces that have knots or "unattractive" features that wouldn't look good under a clear or transparent/translucent finish. In the strictest sense, that can have an effect on the tonal character of the instrument, but that effect would quite likely be so small you'd need instrumentation from NASA to detect it.
I would be more concerned about the neck pocket dimensions. For reasons that have never been revealed, many low-end guitars take a lotta license with those numbers, so trying to fit a "standard" neck might present some challenges. Oddly enough, Fender set those standards and they're the worst violators of it. Warmoth follows Fender specs better than Fender does. Go figure.
If you're gonna replace everything but the body, you might consider just going ahead with the project. Buy all the parts, and before ripping the existing Squier to pieces just pull the neck and see if the new one will fit. If it turns out it won't seat right, used bodies are about a dime a dozen on Ebay. Put the existing neck back on the Squier and flip it to help pay for the "new" partscaster.