Should I run my salt chlorine generator when the water is below 60°F?

No. Running it in cold water will damage the cell and waste your time. The electrolysis reaction slows down below 60°F, producing almost no chlorine while scaling up the plates and shortening the cell’s life. Most manufacturers say not to run it below 50–60°F, and some will void your warranty. Leave the pump on for circulation but turn the generator off, and add manual chlorine until the water warms up.