DS3231 RTC module SQW pin: High/low for extended time?

Here's the schematic of the ZS-042 module showing what I would change. As you can see, if there's no power coming in on Vcc, there won't be any pullup on INT/SQW. So you need to cut that trace to prevent it from becoming a pulldown resistor. Then you can either use INT/SQW to turn on a mosfet, which will have its own pullup resistor, or you can use it to interrupt a sleeping Uno by configuring the Uno GPIO pin as INPUT_PULLUP. That way you can turn off the Vcc power to the module and save battery except when you need to communicate with the module over I2C.