I just fell into this rabbit hole today myself. I couldn't find anything in any newspaper I searched, just Perot's disdain for her design before and after her ethnicity was revealed. Nothing in print seems to show up until after 2000.
But I did see on Maya Lin's Wikipedia entry that the claim is sourced to this book:
Yellow: Race in America Beyond Black and White
Copyrighted Material, The Perpetual Foreigner, page 95 (snippet): [continued from a previous page] a number until her sketches were selected. Once her face was attached to her art, there were murmurings that she was the wrong choice because she was a "gook." For example, businessman Ross Perot, a major promoter of the project, frequently called Lin "eggroll" and, according to press accounts, "he hated that she was Asian."28 Although her monument has become the most popular tourist attraction in the nation's capital, bringing together veterans, protesters, and families, who make crayon rubbings of their loved ones' names, the reticent sculptor still expresses shock at the attempts to discredit [end of snippet]
(snippet of footnotes for above passage): [omitting 26-27, which are not referenced in above passage] 28. Sidney Blumenthal, "He's Ba-a-ack!," New Republic, October 19, 1992, 14. [omitting 29-30, which are not referenced in above passage]
So it's from a Sidney Blumenthal article "He's Ba-a-ck!" written for the New Republic October 1992 issue. The fact it's seemingly originating from Sidney Blumenthal (a known Clinton associate) right during the 1992 presidential election really adds a strange wrinkle to this. As does this sole source popping up nearly 10 years after the incident.
Unfortunately, I wasn't able to see this article. A website that archives periodicals had this issue but blocked PDF download due to copyright. It's also mentioned in some academic pages but blocked behind an EDU login. Curiously, it's not on New Republic's archive on their website (like other articles by Sidney from 1992 are).
So far I'd say the claim is dubious until a proper source is found. This could turn out to be it and I hope someone here can see that article to check. It's strange that no one from the time has any account of it beforehand that I was able to find.
Edit by Avery
Here is the exact quotation; bold emphasis is mine:
But there was another reason that Perot was determined to destroy the relationship. "You don't think that I'd let my daughter marry a Jew," he maintained time and again to a man he trusted. The source tells more: "The notion his daughter would marry a Jew drove him absolutely wild. He wasn't embarrassed talking about it. He'd say, 'You need to have your grandchildren look the way you do.' "
Perot's bigotry is part of his larger paranoia. Whenever he senses a challenge, real or imaginary, he seeks out an object for his contempt. Consider the Vietnam Memorial. The veterans who sponsored the design he despised were investigated and hounded by detectives secretly run out of the office of Roy Cohn; in private, meanwhile, Perot frequently denigrated the winner of the design competition, Maya Lin, as "Eggroll." "He hated that she was Asian," said a source to whom Perot mentioned his nickname for her several times.
Sidney Blumenthal, "He's ba-a-ck!" (Oct 19, 1992). The New Republic, 207, 12. Retrieved from https://search.proquest.com/docview/212862473
Blumenthal also references denials of his previous articles:
Many of the bizarre stories about his obsessive privacy and paranoia -- hotly denied in his previous incarnation -- have been amplified and confirmed in the interim.
TNR was an anti-Perot publication and published an editorial "The Present Danger of Ross Perot" on 29 June 1992. On the other hand, it is known that Perot was angry at the design of the Vietnam memorial. Todd Mason's Perot: An Unauthorized Biography (Homewood, IL: Dow Jones-Irwin, 1990), pp.166-198, does not mention the slur but it does discuss the designer.