How to turn off DC-DC step-up converter to run on 3 AA batteries
Working at such low voltages is a bit of a challenge, I'd suggest first of all that the ULN 2003 is replaced with a TPL7407l (http://www.ti.com/lit/ds/symlink/tpl7407l.pdf) this will get your saturation voltage down to a more acceptable level for driving the stepper motor.

To switch the boost convertor off without having to figure out adding wires to the module you could add a high side switch using a P-Channel switch such as the MIC94052 (http://www.ti.com/lit/ds/symlink/tpl7407l.pdf), this is capable of 2 A with reasonably low RDS(on). If it still proved too high an RDS(on), you could just parallel 2 of them. 
When your ATTiny goes to sleep, all the outputs will float and leakage current for the device is under 5 uA.