Because I'm a proud employee of Horowitz Cabinets and Paneling!
Of all the religious Right's suspicious bumperstickers, this one is the most deserving of scrutiny. Today's Christian radical strikes a bizarre balance between philo-Semitism and anti-Semitism, supporting the existence of the State of Israel only insofar as it leads to the Battle of Armageddon and the death or conversion of the Jewish people; despite their misguided and semimagical means, their wished-for end is still the annihilation of the Jews.
Consider the statement, "My boss is a Jewish carpenter!" The exclamation point must be considered integral to the text, as to my knowledge it always appears on the bumper stickers. The statements on bumper stickers invariably suggest a context: typically this context is the implied or presumed sentiment or statement of the observer, to which the bumper sticker functions as a retort: "My other car is a. . .", in its unironic manifestations, is a response to an implied or predicted "What a lousy car." "You wear your X, I'll wear mine" follows from the viewer's presumed shock or offense at seeing racist symbols prominently displayed, and makes the further assumption that the viewer thus offended is African-American and in possession of mid-1990s Malcolm X movie paraphernalia. Some bumper stickers explicitly state the sentiment to which they later respond: "Don't like my driving? Dial 1-800-EAT-SHIT".
Therefore it is not unreasonable to consider the range of prompts which elicit the response "My Boss Is A Jewish Carpenter!" In the context of early-21st century America, it is unlikely that this constitutes some sort of brag: Jews are not stereotypically associated with woodworking, and those who are inclined to associate the notion "Boss" with the notion "Jew", that is, anti-Semites, would not be proud of their position.
There is likewise no tradition of comparing one's boss to other bosses on the basis of ethnicity so far as pride or prestige is concerned. The use of emphasis, i.e. "My Boss Is A Jewish Carpenter", would tend to imply an unspoken "And Yours Isn't"; no such emphasis is found in practice.
There is only one common context which inclines toward the elicitation of responses such that "ACQUAINTANCE is a MEMBER OF ETHNIC GROUP": the challenge of another person's racial politics, or the challenge of a person's qualification to make an assessment about another ethnic group. That is, "My Boss Is A Jewish Carpenter!" is functionally analogous to statements like "Some of my best friends are Black," and "My son-in-law is Hispanic". Commonly such statements are followed or preceded with the qualifying conjunction "but".
The claim, then, that "My Boss Is A Jewish Carpenter!" only makes sense under the logic of bumper stickers if the displayer is held to presume that other individuals could reasonably have cause to doubt the displayer's belief in the moral worth of Jews. If the displayer did not expect that facts which he or she believed to be evident would lead to the conclusion that the displayer was an anti-Semite, there would be no reason for the sticker in the first place.
The sticker as displayed unironically is ironically never a statement of objective truth; it instead serves to identify the bearer as a Christian, whose deity is putatively assumed to have some training in carpentry as a result of his parentage. Profession of Christianity is therefore used as a means to definitively deny anti-Semitic leanings when in fact history has shown that the preponderance of anti-Semitic activity in Eurasia and North America since the fifth century AD was carried out by Christians.
There can of course be ironic uses of the Jewish Carpenter sticker that do not fall under discussion here; these are paradoxically limited to all cases in which the displayer's actual employer is Jewish, and one's actual profession is woodworking.