No. Rust is iron corroding, and 316L surgical stainless steel is specifically alloyed (with chromium and molybdenum) to resist corrosion — it’s the grade trusted in marine hardware and medical tools for exactly that reason.
With normal everyday wear — including water, sweat, and humidity — your Stylr pieces won’t rust. (After ocean swims, a quick fresh-water rinse is still a good habit for any jewelry.)