Should employers be allowed by law to add a gender preference searching for an employee for any job?

We have enough issues with sexism in the workplace, we don't need to add anything more to make it easier.

What if an employer is looking for a female model? Of course they are going to prefer women as an employee. An employer is a person, and in our society, has a constitutional right to be as discriminatory as they want to be unless they are a public institution. If we feel that the discrimination is unjust (i.e., preferring one gender over the other in a job where that has no bearing), then it is the duty of us, the people, to refuse to support that business until they change their ways. Now, in certain cases, where there is a history of unjust, institutional discrimination, the intervention of the government in private affairs can be justified (i.e. the civil rights movement of the 60s). I think that our society (in the US) is able to overcome this issue without the government interfering, although that is a more subjective argument, which is the subject of debate today.

The only situation where I can see gender preference being relevant are jobs that exploit (mostly) women. When heterosexual men go to a strip club, they don't want to see a dude on stage. Likewise, when heterosexual women go to see the Chippendale dancers, they don't want to see a woman or women on stage. Then there's restaurants like Hooters and The Tilted Kilt that exploit young women's looks to attract male customers. I'm not sure if there are similar restaurants exploiting young men's looks, but I wouldn't be surprised.

I guess if you think those sort of businesses should be allowed to exist, then employers hiring people to work in such places should be allowed to add a gender preference when looking for employees, because gender (and appearance) clearly matter in the performance of those jobs. Come to think of it, if you're casting for a movie/play/t.v. show, you would hire according to gender too. George Clooney wouldn't make for a very convincing Amelia Earhart for example. But in jobs where a person of either gender is not part of what qualifies one to do the job, I would say no, employers should not be allowed by law to add a gender preference when searching for an employee.

Yes, within very strict bounds.

Specifically, the only justifiable reason I can think of is as an affirmative action policy. If it is clear that a woman's apparent qualifications on paper do not represent her actual ability in the field due to institutional sexism having put barriers in her achievement trajectory, it is actually unjust and unfair to not take that into account. It's literally as unfair to not take into account someone's race or gender or class or geography when adjudicating their qualifications as it is to not take into account their volunteer work.

Aside from that, it should not be allowed to hire more young girls than boys and men to have a "pretty" workstaff, or to hire more women because they are ostensibly on average more empathetic, etc. It would be reasonable to want to have a good balance of men and women in some service capacity: For example, social work organizations should probably make sure to have at least 25% of women and men because it is a relevant metric for service.

No not for ANY job, with with certain jobs they already can, with limitations. The exemptions are basically where something is gender specific, in acting you can say this part is for women or for men only. But unless the job is gender specific that is where the other gender CAN'T do the job then the job should be open to anyone at the same pay rate.