While you absolutely can use whatever you have to take class, let me explain WHY you may want to invest in jumpstretch bands and how they are different from the therabands/stretchy bands used in Pilates and yoga classes.
Jumpstretch bands are the same bands that you may have been seen used for assisted pull-ups. They are commonly used in the strength and conditioning space. Jumpstretch bands have a different level of tension and offer more feedback to the muscles than a stretchy band.
Generally speaking, jumpstretch bands are a tool that was developed specifically for increasing strength and research has even suggested that they are comparable to using weights, so while there is nothing wrong with a stretchy band, it has a different application. Stretchy bands are great for building stability and body awareness, but they're not the best tool for strength.
Finally, in many ways jumpstretch bands are an ideal option for strength training at-home, because there are lots of resistance options, they are inexpensive, easy to store, and they're a great place to get started with strength if you are hypermobile or have some mobility issues, because they can be easier to navigate than weights.
This was all a long winded way of saying use what you have, but you will probably find you get stronger faster and get a lot more out of the strength classes if you get the bands.