the explanation given for ゆで卵 claims that the ゆ comes from 湯 - hot water. however when looking on e.g. jisho.org, they give an alternative spelling of 茹で卵, with the 茹 meaning “boil”.
even if 茹 is a rarer kanji, and perhaps not covered on WK, it seems wrong to give what seems to be an incorrect explanation. in particular on a kanji learning website.
Joking aside, Wiktionary concurs that it’s from 茹でる, though it’s lacking the etymology that it often provides. 湯 doesn’t put in any appearence at all, either way.
Goo has ゆでる as a sound change of うでる (and another kun’yomi for 茹 is う.だる), so unless 湯 also changed from う to ゆ at some point and Goo doesn’t have it bc it’s never read う anymore, I’m not so sure they are related linguistically despite being so conceptually
Also, it doesn’t have etymology, but it does have two different ways to write it in kanji: