TLDR: Advertise your candidates of all backgrounds as good as possible
Preamble: As this seems a bit unclear and differently interpreted, I read the question such that the white males are filtered out before even looking at their profile and as such without basing the decision in any way on their qualification but just their skin colour/ethnicity.
Your job is two-fold: Find a matching candidate for a company and find a company for your candidates. While typically the companies pay you (your agency), sometimes candidates pay too and even if not, the more of them you can get a job the more companies will pay you. So in that sense, you work for both. So your job is to look out for both. (On top of that we obviously all want to be good people and not support discrimination but rather contribute to a fair society.)
There are a few good business reasons to hire from certain demographics or ethnicities. Kate Gregory pointed out several in her answer and the comments. For example, companies might want to sell their products to as many people as possible. For this it can help to understand different communities and how best to address them, e.g. via advertisements etc. Sometimes having an employee from certain communities/ethnic backgrounds can help in this. Same for the design of products, sometimes there are oversights when products are developed in-house just by members with one particular background (e.g. there are apparently soap dispensers that do not recognize black hands). Another company might want to sell to shops in China Town, but these mainly trade with other Asians (the example is made up, no idea if that is the case to any degree). In that case it might help to hire an Asian in the hope they can easier get connections into this community. Sometimes a certain background can help to prevent PR pitfalls, e.g. use a historical figure in your video game or as a company advertisement character etc. but overlooking their troubled past regarding slavery. Having a black person in the team could have helped to bring that up. On the other hand it might also just be a PR measure to increase the diversity rate and project an image of progressiveness in order to improve the company's standing with the general public (and attract interest and perhaps customers).
However, in many cases having a certain ethnic background does not guarantee a candidate can automatically help in whatever goal the company wants to achieve. And in particular, another person might have a unique background that helps achieve that goal better.
For instance, a white guy with a literature degree might also point out H.P. Lovecraft could ruffle some feathers when used to represent your company, perhaps more likely than a random black person (that otherwise is equally qualified for your designer/management job). Or a black woman that already worked with a lot of Chinese companies might also know how to get a foot into your Chinatown businesses etc. rather than a random Asian person. It comes always down to evaluating the individual as a whole to find the best candidate. In particuluar, if race or gender is used as a rough brush to filter out candidates before looking at the individual candidates this process may overlook all around better qualified candidates and we get indeed in the hot water of (potentially illegal) discrimination (see below).
So the first thing you should do is talk to the companies/departments that request candidates of certain demographics and clarify their underlying goals. Perhaps you can then add a few white males that would help them in achieving their goal too.
Now, given the current political climate, I'd consider it likely that at least a good portion will just answer that they want to increase their diversity, without having actually looked at any business benefit from that for the concrete position they are hiring for. Likely this is more due to the general goal of 'being more diverse'. They might feel that it has an inherent benefit in general or they might want to incorporate that in their PR measures. Perhaps their leadership (and perhaps good parts of their employees) just feel that this is a way to right a wrong in society as a whole.
I'd still argue to try and dig as deep as possible without becoming obnoxious in these cases. First to find out what kind of candidates they actually consider a diversity fit for them (what about a trans-person? what about a "hillbilly" in their totally progressive mono-universe^^). And second to make them think a bit about their own position and perhaps rethink it.
Especially in these case, if you feel some white males would otherwise be better qualified, you might want to advertise for them too and send a mail along the lines of "here are the candidates filtered by your criteria. Please note that we have some otherwise excellent candidates matching all criteria except 4.b (where 4.b is them not being of the target ethnicity).
Still, some will just blindly go for whatever they or their bosses consider diverse. Depending on how they do that and in which state you are in that might be illegal. You might want to clarify the legality by asking on law.stackexchange.com or by talking to a lawyer. In cases of obvious illegal practices, you could then also hint at the practice being illegal according to law so and so to them or anonymously report these practices. Though you'd need to be very careful in that regard. Even if in the end you might be right in court, you could loose your job. So I'd make sure first where the legal boundary is drawn and start playing it safe, not saying that a certain requirement from a company is illegal but just indicating the law and indicating that they might want to check with their own legal department whether that process is legal for them to apply. Depending on your boss and how strong you feel they are supporting such practices, you might want to get their backing for such cases first or try to keep this between you and the customer (the latter obviously has more risk if the boss would otherwise agree with you). For a rough guideline of what behaviour is legal, you might also checkout this link: https://www.kcsourcelink.com/blog/post/blog/2018/05/01/how-to-recruit-and-hire-for-diversity-legally#:~:text=Intentionally%20recruiting%20for%20diversity%20is,not%20to%20violate%20antidiscrimination%20laws
(provided by @Joe W in a comment)
In the end, how much you need the job and how strongly you feel about such discriminative* behaviour only you can decide - and thus only you can decide how much weight you want to put into questioning the practice. You obviously always have the option to look for another job to not support this, but as you already know, that has it's personal drawbacks (i.e. one needs to find a job first or is without money^^).
* Is it discriminative (or even racist):
While there are positions where ethnicity can be an indicator for a better fit due to the background, I'd wager for most cases in the US at the current time this is not the case and it is "just" about increasing diversity. I find it personally okay to pick a candidate for better cultural fit (after taking all the other qualifications into consideration). An ethnicity alone is not sufficient to a priori indicate that though. I also find it okay to try and counter-balance any subconscious bias by going for a minority by default if candidates are otherwise equally qualified. Whether they are equally qualified, however, is only known once the candidates have been evaluated individually. So if a company asks to filter out certain groups based on gender/sex and/or race before even looking at the candidates, I find there is no way around calling it discriminatory. In the first case where there is at least a business case to be made, I'd consider it a mild case, while the latter is not. However, in a case like this (as a measure to counter existing imbalances) it is not racist in the stricter sense that people associate one race with worse attributes (while still being racist in the wider sense as in "decision based solely/primarily on race"). It would still be discriminatory, morally wrong and potentially illegal depending on local legislation (but please clarify with a lawyer if you feel it falls into that category).