This article says they vary a lot. I say a Slushie is a Slushie, if it is not carbonated. Or no vodka is added.
A slushie is just a snow cone you can drink. 
I don’t touch the stuff but I did work on the machine. Shortly after high school I worked at a coffee shop that offered granitas. That term implies a higher-end product but these were, in fact, just slushies. Interestingly, the temperature control was completely mechanical. The freezer would kick on while the spiral mixer never stopped. Once the ice content got to a certain point, the torque the motor had to exert to keep the mixer spinning would cause the whole mixer assembly to pivot along the axis of the shaft which opened a switch and killed power to the freezer. The position of the switch was adjustable so you could make it a little more or less icy. However, the whole thing frequently froze up and an intentionally weakened shear part would act as a mechanical fuse and make the whole thing stop working. This happened a lot and we eventually got rid of it.
I once went to a party where the hosts had rented a Slush Puppie machine and were mixing them with vodka. A great idea. But one seemingly much copied by local drunkards the few times I made it to the beach in Vancouver.
I am sure carbonating slushie has an effect on texture. Never really noticed the 7-11 slurpees were carbonated.
I have never had squishee made entirely of syrup.