Mazal tov on the birth of your baby!
Haakaa makes several different devices, and other companies make comparable ones. We’ll discuss the halachot that apply to each type of device that might be relevant here.
Milk collector or shell
One device is a milk collector (Haakaa) or milk collection shell (Medela). These products apply no suction, but merely collect milk from the non-nursing breast. Since there is no action of expressing or drawing out milk, we follow the view that milk collected in one of these devices is halachically permissible for use after Shabbat.
Sometimes, a milk collector can be applied to the breast in a manner that creates very light suction, which can increase the amount of milk collected. If a device can be used with different levels of suction, it should be used on Shabbat with the lowest level of suction that will keep it in place and, if necessary, relieve engorgement. This is still not considered active pumping, and the milk is halachically permissible for use after Shabbat, as above.
Suction pump
Haakaa also makes a suction pump, which draws milk from one breast more actively while the baby nurses from the other, still using suction. If you can set this type of pump to very light suction, so that it more closely resembles a milk collector or collection shell, it would also be permissible and you could use the milk after Shabbat, as above. The collector or shell would be preferred.
However, if you use a suction pump of this sort on full suction, then placing it and activating the suction is akin to pumping. That would be permissible if needed to relieve the engorgement, but in this case you should not keep the milk.
Medical note
Please note that to ensure the medical safety of drip milk from the second breast, it should be refrigerated or frozen within an hour of collection, and the shell and breast should be cleaned before the shell is used while feeding from the other breast.
We hope this is clear. Please get back to us with any follow up questions.