I have no idea where that notion came from. I am reasonably sure that a signature in red ink, or green ink, or any other unusual color, is just as binding as one in blue or black ink. Indeed a signature need not be in ink. Blood is actually somewhat traditional.
I suppose some shades of red might not photocopy well.
§ 3-401 of the Uniform Commercial Code (UCC) reads:
(b) A signature may be made (i) manually or by means of a device or machine, and (ii) by the use of any name, including a trade or assumed name, or by a word, mark, or symbol executed or adopted by a person with present intention to authenticate a writing.
§ 3-402 reads:
(a) If a person acting, or purporting to act, as a representative signs an instrument by signing either the name of the represented person or the name of the signer, the represented person is bound by the signature to the same extent the represented person would be bound if the signature were on a simple contract. If the represented person is bound, the signature of the representative is the "authorized signature of the represented person" and the represented person is liable on the instrument, whether or not identified in the instrument.
However § 3-406(a) reads:
(a) A person whose failure to exercise ordinary care substantially contributes to an alteration of an instrument or to the making of a forged signature on an instrument is precluded from asserting the alteration or the forgery against a person who, in good faith, pays the instrument or takes it for value or for collection.
If it is somehow easier to forge or alter a signature in red ink, 3-406(a) might apply, but I don't see why that would be.
But there is no specific mention of color of signatures in the UCC.
Under 15 USC 7001 and subsequent sections, electyronic signatures must generally be accepted in commerce.
(a) In general
Notwithstanding any statute, regulation, or other rule of law (other than this subchapter and subchapter II), with respect to any transaction in or affecting interstate or foreign commerce—
(1) a signature, contract, or other record relating to such transaction may not be denied legal effect, validity, or enforceability solely because it is in electronic form; and
(2) a contract relating to such transaction may not be denied legal effect, validity, or enforceability solely because an electronic signature or electronic record was used in its formation.
Upcounsel's page on "Legally Binding Signature" states:
What Constitutes a Signature?
A signature may be issued by anything that marks on paper. The pencil is not the ideal choice because it can erase or be smudged, but signatures made in pencil are just as valid as signatures based in pen. Signatures can be issued in digital form or via stamps because there are various forms of writing implementations. If you cannot sign an agreement on your own, you can give it to another party who can sign documents on your behalf. You may also use what’s called a digital signature, a way of signing documents that’s not in printed form.
FindLaw's page on "What Are the Rules Regarding Signatures in Contracts?" states:
Usually, a signature is simply someone's name written in a stylized fashion. However, that is not really necessary. All that needs to be there is some mark that represents you. It can be -- as many signatures end up -- a series of squiggles, a picture, or historically, even the traditional "X" for people who couldn't read and write. As long as it adequately records the intent of the parties involved in a contractual agreement, it's considered a valid signature.
Usually this mark is made by a pen, but not necessarily. The signature can be made by anything that marks the paper. Pencil is not favored because it can smudge and be erased, but a signature made with a pencil is equally valid as a signature in pen. Signatures can also be made with stamps or with electronic means, since these are all different forms of writing implements.