Like when people who receive free government aid, and complain about regulations on it. Not what the person would be called, but what it is called that they're doing.
Don't question the value of a gift. The proverb refers to the practice of evaluating the age of a horse by looking at its teeth. This practice is also the source of the expression “long in the tooth,” meaning old.
Not one word, but I would call this biting the hand that feeds you.
The Free Dictionary:
bite the hand that feeds you: to severely criticize the person or organization that helps you or pays you
Ungrateful is just about the closest thing to a one-word solution that I can think of. You would call them ungrateful and you would say that they are being ungrateful. Of course, ungrateful describes many other situations than just the one you're asking about; there are many more ways to be ungrateful than just complaining about something that was given to you for free. Which is likely why the idiom "to look a gift horse in the mouth" came about.
You look in a horse's mouth to determine whether or not the horse is healthy. So, when given the gift of a horse (which would be akin to being given an automobile today), looking it in the mouth instead of just accepting the gift would be considered rude as it would be a show of ingratitude, as if you might not accept the gift if it isn't good enough.
Looking a gift horse in the mouth is not quite the same thing as complaining about regulations on free government aid. They are both phrases that describe an act which would be considered ungrateful, but the first because one does not refuse a gift and the second because the aid was requested; it is not so much a gift as a FAVOR and it is considered ungrateful to complain about that at ALL.
Hope this helps.
Such people are just being unappreciative - not appreciating the value that something provides (and hence nitpicking on other, even minor aspects).
Not fully understanding, recognizing, or valuing something
‘So often football supporters are accused of being unforgiving and unappreciative.’