An employer can require any arbitrary employee to wear a bra, or to not wear one, unless there is a overriding contractual clause (a freedom-to-dress clause), or some statutory limit. Such limits can arise from municipal law, state law, and federal law. It is obviously impossible to review all applicable municipal and state laws, which would typically involve the same mechanism as the federal rules.
The underlying federal law pertaining to sex and employment is 42 USC 2000e–2 which most saliently says it is illegal for covered employers
to fail or refuse to hire or to discharge any individual, or otherwise
to discriminate against any individual with respect to his
compensation, terms, conditions, or privileges of employment, because
of such individual’s race, color, religion, sex, or national origin
("Covered" employers are those with 15+ employees for at least a cerrtain period of time).
The bottom-line regulations and hints as to what the law is are in 29 CFR part 1604. There are bona fide occupational qualifications which can allow hiring only males or only females, but they are narrowly construed (can't be based on stereotypes like "females are weaker" so decisions must be based on individual characteristics). There are no specific rules against systematically mistreating females (and not males) in employment, and there is no clear definition of what constitutes mistreatment.
The law does not prohibit restrictions on appearance, and does not limit appearance restrictions only in case of "dealing with the public". The law does allow general rules rather than case-by-case arbitrary decisions, indeed encourages general rules over subjective determinations ("lack of a bra is not immediately obvious"). The rules do not have to be identical for males and females: see Kleinsorge v. Eyeland, 251 F.3d 153, where there were different rules for men vs. women regarding wearing earrings. See cases cited therein, including Knott v.Missouri Pac. Ry. Co., 527 F.2d 1249 ("minor differences in personal appearances regulations that reflect customary modes of grooming do not constitute sex discrimination within the meaning of §2000e-2") and others. See also Jespersen v. Harrah's, 444 F.3d 1104, which held that
appearance standards, including makeup requirements, may well be the
subject of a Title VII claim for sexual stereotyping, but that on this
record Jespersen has failed to create any triable issue of fact that
the challenged policy was part of a policy motivated by sex
stereotyping
In Gerdom v. Continental, 692 F.2d 602, the issue was weight requirements imposed on females but not males, and Continental was found to have illegally discriminated because "it merely declares the company's determination to impose this requirement on employees because they are females", also noting that "gender-based discrimination cannot be upheld on the basis of customer preferences unrelated to abilities to perform the job".
The basic logic of such a case is that a plaintiff could allege sex discrimination in a practice, then the defendant would have to offer a non-discriminatory rationale. If the rationale seemed credible (that is, the reason was something other than just "we want females to wear bras"), the defendant would probably prevail.
As to the question of whether anonymous employee complaints could constitute a sufficient rationale, that is hard to judge in lieu of case law, but I doubt that such an argument would fly. Such a rationale could possibly address a legitimate interest of a business. Issues of employee morale or distraction and hostile workplace are legitimate business concerns, but a single anonymous complaint would hardly establish that there is a real problem for the business to address. A massive outcry from the workforce could make the concern look more credible, especially if it were based on a "deeply held value" of said employees, and there was a believable threat that a substantial chunk of the workforce would leave if the situation was not remedied. An anonymous complaint (or multiple complaints possibly from the same person) would not make such a "threat" credible.
One limit on an employers legitimate business interests is that they must give way to making reasonable accommodation to an employee's religious requirements (EEOC v. Abercrombie & Fitch). If an employee objected to working in the same room as a braless female, for some religious reason, then reasonable accommodation to that consideration is necessary. The most reasonable accommodation is allowing the offended employee to work in a different room. Things get more complicated if, for some reason, that accomodation is impossible or highly impractical. In order to successfully invoke the religion card, a complainant must provide evidence of an actual religious need. In Swartzentruber v. Gunite Corp., 99 F.Supp.2d 976, plaintiff wished to argue that he had a religious right to display KKK tattoos such as the "Firey Cross". But,
Swartzentruber does not present admissible evidence, or even contend
without evidence, that being required to cover up his tattoo at work
conflicts with his religious beliefs, or that he told Gunite about any
conflict with his beliefs and Gunite's demand that he cover his tattoo
Even if the employer had been put on notice about the religious claim and the complainant had substantiated the claim at trial, that does not give an emplyee a right to do whatever they want:
the court agrees with Gunite that any greater accommodation would
cause it an undue hardship. Gunite demanded that Mr. Swartzentruber
cover his tattoo because it violated Gunite's racial harassment policy
and offended other employees. Gunite accommodated his tattoo depiction
of his religious belief that many would view as a racist and violent
symbol by allowing him to work with the tattoo covered; Title VII
doesn't require more.
