DS1307-breakout-board
Those resistors on SDA and SCL are I2C pullups. In order to work I2C bus needs proper pullups. This is because the IOs on the bus are open drain.

Here is an excerpt from i2c-bus.org:

The I2C bus transmits data and clock with SDA and SCL. First thing to realize: SDA and SCL are open-drain (also known as open-collector in the TTL world), that is I2C master and slave devices can only drive these lines low or leave them open. The termination resistor Rp pulls the line up to Vcc if no I2C device is pulling it down. This allows for features like concurrent operation of more than one I2C master (if they are multi-master capable) or stretching (slaves can slow down communication by holding down SCL).

Best place that explains SQW out is the datasheet. In the ref design below, the pullup is there on the SDA, SCL and SQW out.


Following is the block diagram depicting the SQW output stage.



Also see the description of the SQW pin mentioning the requirement of a pullup resistor.