"Can Adria be fired for stating publicly ..."

It depends on several factors. A company cannot use a legitimate reason for firing someone as a pretext for an illegal reason to fire someone. The courts will ask questions like: Does the company have a policy that if an employee makes an incorrect statement like this then that person can be fired? Are there examples of other employees who made an incorrect statement but did not get fired? Here's the actual EEOC statement:

> The anti-retaliation provisions make it unlawful to discriminate
against any individual because s/he has made a charge, testified,
assisted, or participated in any manner in an investigation, proceeding,
hearing, or litigation under Title VII ...

> The anti-discrimination statutes do not limit or condition in any
way the protection against retaliation for participating in the charge
process. While the opposition clause applies only to those who protest
practices that they reasonably and in good faith believe are unlawful, the
participation clause applies to all individuals who participate in the
statutory complaint process. Thus, courts have consistently held that a
respondent is liable for retaliating against an individual for filing an
EEOC charge regardless of the validity or reasonableness of the charge\26.
To permit an employer to retaliate against a charging party based on its
unilateral determination that the charge was unreasonable or otherwise
unjustified would chill the rights of all individuals protected by the
anti-discrimination statutes.

> Typically, pretext is proved through evidence that the respondent treated the complainant differently from similarly situated employees or that the respondent's explanation for the adverse action is not believable. Pretext can also be shown if the respondent subjected the charging party's work performance to heightened scrutiny after she engaged in protected activity.

"If I have an employee who ..."

The law gives broad anti-retaliatory protections to someone who made a complaint. As I quoted above, the courts protect a claim made in good faith. "To permit an employer to retaliate against a charging party based on its unilateral determination that the charge was unreasonable or otherwise unjustified would chill the rights of all individuals protected by the
anti-discrimination statutes."

"Good faith" is given a very broad interpretation. It's not expected that an employee will know the Title VII laws, nor know if a certain practice is actually prohibited under the law. However, if it was not made in good faith (the wording is something like "no reasonable person would agree"), then that complaint does not have Title VII protections.

Also, the employee may not make unreasonable forms of complaint. The EEOC writes: "[C]ourts have found that the following activities were not reasonable and thus not protected: searching and photocopying confidential documents relating to alleged ADEA discrimination and showing them to co-workers\17; making an overwhelming number of complaints based on unsupported allegations and bypassing the chain of command in bringing the complaints\18; and badgering a subordinate employee to give a witness statement in support of an EEOC charge and attempting to coerce her to change her statement.\19 Similarly, unlawful activities, such as acts or threats of violence to life or property, are not protected. ... If an employee's protests against allegedly discriminatory employment practices interfere with job performance to the extent that they render him or her ineffective in the job, the retaliation provisions do not immunize the worker from appropriate discipline or discharge."

That tweet can be parsed in many ways. Does "support me" mean that the company agrees that hearing unwanted dirty jokes while at the workplace is a valid reason to complain? Does "support me" mean that the company supports her legitimate use of Title VII protections? Does "support me" mean that her supervisor told her she was supported by the company, but upper-level management later decided that she wasn't?

Given that a company is supposed to support an employee's Title VII civil rights, I think that there is a reasonable person (the law says 'a reasonable person' not 'every reasonable person') who would say that tweet is nowhere near the bounds of what is impermissible in a Title VII action.