Such claims might well constitute False Advertising.
In the US state of New Mexico, False Advertising is prohibited by NM Code section 57-15-1 which says that:
False advertising in the conduct of any business, trade or commerce or in the furnishing of any service in this state is hereby declared unlawful.
Section 57-15-2 defines false advertising as:
... advertising, including labeling, which is misleading in any material respect; and in determining whether any advertising is misleading, there shall be taken into account (among other things) not only representations made by statement, word, design, device, sound or any combination thereof, but also the extent to which the advertising fails to reveal facts material in the light of such representations with respect to the commodity to which the advertising relates under the conditions prescribed in said advertisement, or under such conditions as are customary or usual.
If a company makes claims about its products or services, and those claims are false or misleading, and they are significant to the decision of reasonable person whether or not to buy the product or service, then they would seem to be false advertising under 57-15-2.
Under section 57-15-4 a business can be subject to a civil penalty for false advertising. But a suit to impose such a penalty can only be brought by
... the attorney general or, with his consent, the district attorney of the district where the act is committed.
Thus, an ordinary consumer may not bring such a suit. However, under section 57-15-5 an action for an injunction may be brought:
A. The attorney general of the state of New Mexico or the district attorney of the district in which the violation occurs or a private citizen may bring an action in the name of the state against any person to restrain and prevent any violation of this act [57-15-1 to 57-15-5, 57-15-9, 57-15-10 NMSA 1978]. Any proceeding initiated under this section by a private citizen shall be initiated on his behalf and all others similarly situated. (Emphasis added)
A private citizen does not obtain any damages from such a suit, but under 57-15-5 (B) in "exceptional cases" such a citizen may be awarded "reasonable attorneys' fees" if s/he wins the suit, and the court finds that the false advertising was "willful".
None of these provisions allows a citizen to simply demand that a company prove its claims, or reveal its business processes. However, in a suit filed under 57-15-5, discovery might reach internal company documents which would help to show the truth or falsity of the advertising.
This page from truthinadvertising.org says that a person may sue in Magistrate's Court (the NM version of Small Claims court) for recovery of the purchase price of something bought because of false advertising. I haven't found a specific law authorizing such a suit. In any case. most small claims courts in the US have no or very limited discovery procedures. According to this page from lawyers.com, the limit of small claims awards in NM is $10,000, and a lawyer is permitted, but not required.