Yes, a son who has "had 3 jobs, all of which he's quit". A son who begrudges you a cheap car because in his eyes you should have "instead spent the money to support him".
So let's pick his comment apart a bit.
In what way does he feel you don't support him? In what way does he feel you giving him money is 'support'? At what age does he feel he is an adult? At what age does he think adults should take responsibility for themselves?
These are questions I think you should actually ask of him.
And - when he visits you at the weekends, what does he actually do? Does he contribute in any way? I'm not talking money, but does he spend time with you, or is he there to visit his friends who are local? Obviously he won't be hoovering and dusting (although he should be responsible for the care of his room) but does he ever offer to do so much as make you a cup of tea while you put your feet up? Or does he expect to still be catered to in the way a child of 10 would be? Does he eat with you and your husband as a family, or does he raid the fridge for whatever he wants and eats in his room or on his way out the door?
As comparison - I didn't expect my son to visit every weekend at 23. He had his own life, his friends, his girlfriend, his job, his studies. He visited maybe monthly, Whatsapping inbetween. (And yes it was often to visit local friends, but we'd eat together and go out for coffee together.)
"My husband is really angry at the comment, saying that my son, as an unemployed person, has no right to tell my husband what he should spend his money he works hard for, on."
Well, your husband is right, and I'm pretty sure you agree with him, don't you? How do you feel about your son's preference that you not have a reliable car but be shovelling money his way instead? I'd be disappointed, and yes - a bit angry.
"He no longer wants to allow my son to come over for weekends, because he says he's not willing to spend money supporting somebody he calls a deadbeat and who has the audacity to tell him, as a working person, how he should spend his money."
I'd have a talk with my husband about my son's visits. Have a think first if your son is visiting you to visit you, or to sponge, or because his dad wants a break from having him all the time. Have a think about what YOU want from your son's visits, and whether or not you're actually getting that. Or maybe your weekends feel as if you're being used as a hotel/crashpad? Have a think. Discuss with your husband what you want, whether you want the visits to continue as they already are, or whether you're dissatisfied with them too in any way and whether you'd like them to change.
And most of all - give your son a ticking off. He's 23 and keeps quitting jobs and expecting his parents to funnel money his way. I'd be upbraiding mine if he pulled that shit on me.