According to this page at Chadrei Charedim, the correct pronunciation (as demanded by the strictures of Hebrew grammar) is with a patach. Indeed, this is true: consider the form of similar phrases that appear throughout Tanakh:
Genesis 7:1 - בַדור הַזה (bador hazeh);
Exodus 5:23 - לָעם הַזה (la'am hazeh);
Leviticus 23:27 - לַחדש הַשביעי הַזה (lachodesh hashevi'i hazeh);
Numbers 20:5 - אל הַמקום הָרע הזַה (el hamaqom hara' hazeh);
Deuteronomy 4:6 - הַגוי הַגדול הַזה (hagoy hagadol hazeh);
In most siddurim, the correct pronunciation is recorded (see, for example, Siddur Vilna). What the Mishna Berurah wrote is therefore puzzling, but is explained by the author of the post at Chadrei Charedim (to which I linked above) as being the result of an unfortunate error.
The assertion that it should be with a chiriq and not a patach was made by the author of the Mateh Moshe on p.296, §980. I have not seen the original text, but it is apparently riddled with other errors of a grammatical nature, as mentioned at Chadrei Charedim. It was subsequently quoted by the Magen Avraham, who mistook it for having been the observation of Rabbi Shlomo Luria (the Maharshal), since the passage in Mateh Moshe in which it appears commences with a quote that is attributed to him.
You can see the Magen Avraham's observation in situ (Orach Chayim 676:1), where he falsely attributes it to the Maharshal. The Mishna Berurah appears to have taken the idea from there, although it is truly mysterious why he would have done so.