As an employer and Veteran, am I within my rights to question an applicant's discharge if they list BCD as reason for leaving?

No, a Bad Conduct Discharge (BCD) is not an administrative discharge - it is a punitive one.

If a former service member* has a BCD, then they means they were usually convicted of a serious offense at a Special or General court martial. I say "usually" as it could be multiple lower-level offenses or even a felony-level if there were mitigating circumstances.

Most often a BCD will have been preceded by time in a military prison. A BCD is a federal conviction and will show up on a background check. The only 'light' spot about a BCD is that it isn't a Dishonorable Discharge (DD). However, you can think of a BCD as "A Dishonorable Discharge's younger brother" as both are characterized as "discharged under dishonorable conditions".

Keep in mind that just like the civilian justice system, a conviction by a court martial* doesn't tell the entire story. An established pattern of not following orders could lead to a BCD as well as everything from indecent language to a minor, Malingering, or missing a movement to assault and DUI.

Although none come to mind, you'll want to make sure that you are compliant with your state EEO* laws especially if your state prohibits/limits hiring decisions based on criminal convictions. While almost all state laws are geared towards the non-punitive discharges (Honorable, General, and Other Than Honorable) I didn’t find any off-hand that protect punitive ones.
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* If someone discharged from the military does not have an "other than dishonorable period of service", then they are not considered a veteran by federal law. However, the individual may be in a 'gray area' with a BCD as its' possible they did have an honorable period of service if they got the BCD after their initial enlistment (VA would have to do a discharge characterization to move them out of the gray area).
* Manual for Courts-Martial of the United States (2019 edition) - https://jsc.defense.gov/Portals/99/Documents/2019%20MCM%20(Final)%20(20190108).pdf?ver=2019-01-11-115724-610
* EEO laws - Of note is that federal laws like USEERA do not apply to individuals who are discharged from the military under dishonorable conditions

A BCD is much more than an admin seperation. BCD’s are generally because of criminal behavior that also resulted in time in the brig in additon to a derogatory discharge status, loss of rank and pay.

All of the BCD guys I know was because they were stealing and selling the goods or some other related shenannigans.

It’s well within your rights to question it because a BCD isn’t just a little bad paperwork.

https://www.military.com/benefits/military-legal/types-of-military-discharge.html/amp

One of the soldiers in my unit in Germany was convicted of stealing his roommate's wallet, stole his blank checks, and found the soldier's ATM PIN and stole $2,000 from him. He received a Bad Conduct Discharge, 3 years confinement, and the other usual stuff. I may be wrong, but if an employer ever asked him if he had ever been convicted of a felony, he would need to answer yes.

As an employer, I believe, you are well within your rights to question an applicant on their discharge from the military. A BCD, depending on the reason, could be a detriment to your business especially if the reason for his BCD conflicts with what your company does. Of course, as always, keep your questions to the applicant professional.

Normally means they got off lucky instead of full court-martial, It is a felony conviction and should cause many questions from any employer.

SN Walt Boyer A BCD can only be awarded by a Special or General Courts Martial. You might want to consult the labor board on this. Personally I believe you have a right to ask him about it. If I was in your shoes I'd ask to see the transcript of the trial. They don't just hand out BCD's like cotton candy. Either this applicant had a bad military history and it was time to dispose of him or the crime was so bad it this was the best course of action. If the discharge was for numerous AWOL's I wouldn't be too worried but you have to decide that for yourself.

This is not a direct answer to your question but still relevant. I have a general discharge under honorable conditions. Almost 20 years ago now I joined in on some hazing which ended my naval career after a captain's mast / NJP. I had an outstanding service record up to that point. Now at 39, I am not even close to the same person I was when I was 21. Unfortunately, no matter what I do, I have had to try and outrun my DD214 ever since. Many jobs will not even look at an application without anything but a standard honorable discharge. Doesn't matter how qualified or how long it has been. All this to say that I understand it is important to look at someone's background, but I am glad you are still considering this person.

You not only have the right, you have an obligation to ensure your client's are protected. A BCD indicates that the SM committed a felony. I would definitely not only ask, but require the potential employee to provide the details of his offense with objective evidence of it as well.