Shulchan Aruch, Orach Chaim 196:1 says that you shouldn't:
אבל [צ"ל אכל] דבר איסור אף על פי שאינו אלא מדרבנן אין מזמנין עליו ואין מברכין עליו לא בתחלה ולא בסוף:
One who ate a forbidden food, even if it's forbidden only Rabbinically, cannot be included in a mezuman [a quorum for Grace After Meals], and does not recite a blessing before or after eating ...
Keep in mind that there are several different categories of non-kosher food. Something like non-kosher beef is a real problem. Similarly, Chometz on Pesach, Yayin Nesech (wine used in idol worship) and meat with milk.
On the other hand, food that is not certified as kosher may indeed be kosher or non-kosher in a (relatively) minor way. If you eat non-kosher ...
No. The Shulchan Aruch, Orach Hayyim 196:1, says:
If one ate a forbidden item (even it is forbidden only by the sages), one cannot make a zimun on it or say a blessing on it, neither before or after it.