Jurors do not swear on a Bible, only those giving testimony before the judge and jury. If the particular case involved upholds an immoral law you need to disqualify yourself. For example, if you needed to uphold a law concerning "rights" to abortion, same-sex "marriage," or euthanasia, you could not be impartial. That being the case, when it comes to the voir dere, you'd be dismissed by one of the lawyers as being biased.
If I'm not mistaken, it is legal in the US to "affirm" rather than swear an oath, so long as you make this known in advance to the judge. Again, this applies mainly to testifying, not jury service.
Since so many people do not want to serve on a jury, the courts now summon everyone, even those formerly excused. A few years ago, my mother was given a "must appear" summons to grand jury located 80 miles from her home. She is handicapped and cannot drive, also, was 91 years of age! So my Dad who was still driving had no choice but to drive her in, request assistance with her wheelchair, etc. They presented the doctor's note and had to be helped to return to the parking garage in order to leave. Four years ago I was summoned along with an elderly man on a walker who was deaf. Despite hearing aids, he could not hear when his name was called. He was dismissed at the end of the day. The next day a woman went into labor in the jury pool room and left via ambulance. She asked to be excused the first day and was denied. Really, the court should be ashamed to require a very visibly pregnant woman to appear.
In a few more humorous incidents in my area, among those receiving "must serve" summons have been the deceased, infants and children, pets with human names, dogs, cats, a parrot, and a horse!
My opinion is that Catholics should serve if they can do so.