It's a kiddush Hashem to keep G-d's law even if people disagree or don't understand. This is the very first law in the Code of Jewish Law: “Do not be embarrassed of the scoffers…” i.e. don’t desist from doing what’s right even if others look askance, disapprove, or are scornful.
It's a chilul Hashem to comply with policies that violate G-d's law, and it's a public chilul Hashem to violate G-d's law in public.
There is a common misconception about "chilul Hashem." It does NOT mean doing something that society disapproves of or deems reckless. There is indeed a concept of "mar’is ayin" – i.e. to avoid doing something that gives people the impression that one is committing a sin  – but that's only when the specific action that the public presumes was committed is a deed specifically prohibited by the Torah.
Likewise, a Torah scholar who committed an action that the public perceives as sinful has desecrated G-d’s Name  only when:
1) that specific action (i.e. taking a purchased item but not paying for it right away, etc.) has objective value that the Torah recognizes as unbecoming, and more importantly
2) it does not itself violate the Torah. For example, many probably argued that Mordechai was causing a "chilul Hashem" for defying the government’s order to bow before Prime Minister Haman, but in reality, the opposite was true.
Mask-wearing is not an objective value in Judaic Law, but just the opposite. Covering one’s face is prohibited by halacha and falls within the rubric of “darkei Emori” – i.e. any man-made ideology, policy, or dogma that’s societally enforced and may not be questioned, since our religion regards such reverential compliance to man-made ideas or policies as idolatrous.
Public policies aside, there are numerous other Judaic problems with wearing masks as mentioned in rabbinic texts. 
There is objective Judaic value to exposing one’s face and not masking it:
When Rabbi Yosef Yitzchak Schneersohn (the previous Rebbe of Lubavitch) was fleeing the Nazis -- may their name be erased -- his son-in-law R. Shmaryahu Gurary suggested he cover his face so he not be seen.
The Rebbe responded: "Ah yid darf zich nisht shemen mit zain tzura -- a Jew should not be ashamed with his face.”
Covering one's face is a chilul Hashem. Covering one's face in public is a public chilul HaShem.
It is a kidush Hashem to expose one’s face. Doing so in public is an act of public kidush Hashem.
Don't be ashamed.
Sanctify G-d's Name and show your face.
 The Talmud derives this principle from the verse “And you shall be clean, from G-d and from Israel.” (Bamidbar 32:22)
 Sefer Hachinuch commandment 295
 Such practices are called darkei emori, “ways of the Amorites.” Leviticus 18:3, Sifra ibid. Shabbat 67a.; Masoret Moshe, page 313; Responsa of the Rashba, 1:413 and Beit Yosef on Yoreh Deah 179:20. “If there is actual curative benefit “רפואה” to the particular custom, then the prohibition of darkei emori doesn’t apply – this means that its curative benefit must be perceivable to a person by natural observation. Otherwise, it is forbidden.”
If a policy’s alleged medical benefit to the individual cannot be unambiguously demonstrated by natural observation – i.e. empirical evidence, or in legal parlance, strict scrutiny – then blind adherence to it is categorically prohibited under the rubric of darkei emori, irrespective of alleged societal benefit.
Masks raise numerous halachic and hashkafic concerns:
a) It is categorically forbidden to wear a mask outdoors on Shabbos unless within an eruv. Even when worn on one’s face, a mask is not considered a garment or an ornament, but a load. [Shulchan Aruch Orach Chaim 301:20. Shabbos 66a; Rif ibid 30b. Or Zarua Hilchos Shabbos 84.]
b) It is forbidden to wear a mask during prayer. [Rabbi Moshe Sternbuch in a recent responsum.] It is not appropriate to stand before a king wearing a mask on one’s face. How much more so before the King of kings, the Holy One blessed be He. Furthermore, one is prohibited from praying whilst holding a load that isn’t for the purpose of prayer. [Shulchan Aruch Orach Chaim 96:1.] A mask is a load according to halacha.
c)One may not wear a mask while teaching or learning Torah. This principle is derived from Moses removing his veil when he taught Torah to the Children of Israel. [Exodus 34:35. Torah Shleima vol.21, pages 179 and 183 in miluim siman 6. Note that Rabbi Shlomo Parchin wrote that Moses’ veil had openings for his eyes, nose, and mouth.] Students need to see the face and mouth of their teacher. This is for the benefit of both the student and the teacher. The world stands on the merit of the breath of children learning Torah that ascends heavenward. A mask blocks this breath.
d) It’s forbidden to get an Aliyah or read from the Torah in a mask. [Torah Shleima ibid cites Sefer Chasidim that early elders would expose their faces during Torah reading.]
e) A Jew is required to greet every individual with a smiling face, בסבר פנים יפות (Avot 1:15). A mask violates this principle by making it impossible.
(f) It's not a Jewish practice to wear a mask. Masks were never mentioned in Tanach, and only appear in halacha for the dubious and discouraged function of frightening children. (Shulchan Aruch O.C. 301:20). Masks are commonplace in pagan worship, and have polytheistic origin. As such, wearing a mask amounts to אביזרייהו דעבודה זרה, and should be discouraged among G-d fearing Jews.
1. In kabala, one’s mouth and nose should never be covered. [Likutei Torah (by Rabbi Yitzchak Luria) states that there is no garment for the mouth or nose. Similarly, Rabbi Shalom DovBer of Lubavtich writes in Sefer Maamorim 5665, that there is no garment for the mouth. Rabbi Moses Cordovero writes in Tomer Devorah that one’s mouth and nose should always remain uncovered, in resemblance of the Divine.]
2. Masks ominously symbolize idolatry, division and estrangement from G-d. [Ohr Hachaim Leviticus 19:4]
3. In esoteric Judaic thought, a person’s face is the tselem Elokim, a reflection of the Divine Face. Placing any barrier or blockage over one’s face is an act of obscuring the tselem Elokim in Whose image the human face was fashioned, and reduces a human being to a faceless droid, as the prophet lamented: “They turned their back to me, and not their face.” [Jeremiah 2:27]
4. Hiding one’s face is reminiscent of G-d hiding His face from us [Deuteronomy 31:17-18]