Csdrhrt

If one makes a bracha on something not Kosher (meat and milk,chicken and milk, pork…) should he eat it to avoid saying G-d's name in vain or not?

He already said God's name in vain since non-kosher food doesn't warrant a blessing. Eating the food now isn't going to help that.

אכל דבר איסור, אף על פי שאינו אסור אלא מדרבנן, אין מזמנין עליו ואין מברכין עליו לא בתחלה ולא בסוף. (שולחן ערוך או"ח סימן קצו:א)‏

If one ate something prohibited, even if it was only prohibited rabbinically, one does not combine him to a zimmun, nor would he say a beginning or after blessing [on that food]. (Shulchan Aruch OC 196:1)

After such a mishap one should say ברוך שם כבוד מלכותו לעולם ועד.

According to the Kitzur Shulchan Aruch (in Laws of Blessing 6:4),¹ in the event of saying a Bracha l’vatalah (an unrequired blessing), the fix is to say, “Baruch Shem kavod Malchuto l’olam va’ed.”

צְרִיכִים לִזָּהֵר שֶׁלֹּא לְבָרֵךְ חַס וְשָׁלוֹם בְּרָכָה לְבַטָּלָה, אוֹ לִגְרֹם לְעַצְמוֹ לְבָרֵךְ בְּרָכָה שֶׁאֵינָהּ צְרִיכָה. וְאִם נִכְשַׁל וּבֵרַךְ בְּרָכָה לְבַטָּלָה, וְכֵן בְּעִנְיַן אַחֵר אִם נִכְשַׁל וְהוֹצִיא שֵׁם שָׁמַיִם לְבַטָּלָה, יֹאמַר אַחֲרֶיהָ, בָּרוּךְ שֵׁם כְּבוֹד מַלְכוּתוֹ לְעוֹלָם וָעֶד.‏ ...‏

One should not consume the non-kosher food. There are a variety of reasons for the various prohibitions against eating all the different types of non-kosher food. Many are enumerated in the Mishnah Torah, Laws of Forbidden Foods. For the various types and reasons, see there.

^{1. See also 10:4 and the sources cited there, particularly the Pri Megadim, for more details.}