Zombies are actually quite problematic, though the problem seems to be with only one female zombie, Kol Haron Api, as Zephania (3:8) said:

כִּי מִשְׁפָּטִי לֶאֱסֹף גּוֹיִם לְקָבְצִי מַמְלָכוֹת, לִשְׁפֹּךְ עֲלֵיהֶם זַעְמִי, כֹּל חֲרוֹן אַפִּי, כִּי בְּאֵשׁ קִנְאָתִי, תֵּאָכֵל כָּל-הָאָרֶץ Because My judgment is to gather the nations to heap upon them My zombie, Kol Haron Api, because My jealousy is on fire, and she [Kol Haron Api] will eat all the earth.

But fundamentally, Jews shouldn't have to worry about the zombie apocalypse, as is made clear in Isaiah (66:14), zombies will only attack G-d's enemies:

וּרְאִיתֶם וְשָׂשׂ לִבְּכֶם, וְעַצְמוֹתֵיכֶם כַּדֶּשֶׁא תִפְרַחְנָה; וְנוֹדְעָה יַד-ה' אֶת-עֲבָדָיו, וְזָעַם אֶת-אֹיְבָיו And when you see this, your heart will rejoice, and your bones will flourish like young grass; and the hand of G-d will be known to His servants, and a zombie [will be known] to His enemies.

However, Isaiah (26:19-20) does offer some practical advice while Kol Haron Api is out and about:

יִחְיוּ מֵתֶיךָ, נְבֵלָתִי יְקוּמוּן; הָקִיצוּ וְרַנְּנוּ שֹׁכְנֵי עָפָר, כִּי טַל אוֹרֹת טַלֶּךָ, וָאָרֶץ, רְפָאִים תַּפִּיל. לֵךְ עַמִּי בֹּא בַחֲדָרֶיךָ, וּסְגֹר דְּלָתְךָ בַּעֲדֶךָ; חֲבִי כִמְעַט-רֶגַע, עַד-יַעֲבָר-זָעַם Your dead will live, my dead bodies will arise; awake and sing, dwellers of ash -- for Your dew is like the dew of light, and Earth, bring to life the ghosts. Come, my people, enter your chambers and shut your doors around you. Hide yourself for a moment until the passing of the zombie.

As far as what to look out for, Shlomo, as usual, has some insight (Mishlei 25:23):

רוּחַ צָפוֹן תְּחוֹלֵל גָּשֶׁם וּפָנִים נִזְעָמִים, לְשׁוֹן סָתֶר A northern wind creates rain, and the faces of the zombified [have] a hidden tongue.

So anyone with a second tongue is likely a zombie.