Clayton "Clay" Louis Hollingsworth is the brother of Blanche, Virginia, Charmaine and Tad Hollingsworth. He was portrayed by Monte Markham in two episodes of The Golden Girls, "Scared Straight" in Season 4 and "Sister of the Bride" in Season 6.
Clayton lived most of his life as a closeted homosexual and hid his sexual orientation from Blanche and the rest of his family. Blanche affectionately calls him "baby brother" because he's the youngest sibling. He was married to an unnamed woman for several years, but this marriage was his attempt to deny his homosexuality until he couldn't hide it anymore.
When Clayton tells Rose that he is gay, she tries to help him to tell Blanche. Afraid to admit that he is a homosexual, Clayton claims that he slept with Rose. Clayton eventually tells Blanche the truth after she insults Rose in a rather hurtful way, and she has difficulty accepting it until she realises during a heart to heart that he is still the same baby brother she has so much in common with.
Clayton visits Miami with his boyfriend, a policeman named Doug and thought out the episode Blanche is uptight and even makes a scene in a public place rather than Clayton say who Doug is. After that scene Clayton reveals he came to Miami to tell Blanche he and Doug are to married she has a difficult time with the news with Clayton telling her off for her reaction, but she eventually accepts it and apologises to the couple.
This theory is from TV Tropes and should be taken as such.
Blanche's reaction to Clayton's coming out could be because of her roots as a Southern woman with religious background and the time the episode aired. At the same time in Series 1 and 2 she shows that she isn't Homophobic at all but maybe it's who Clayton is that effects her reason. Until the Golden Palace, we were to believe that Blanche had Clayton as her only brother but then we are introduced to Tad. Because of his disability, Blanche had been told to treat Tad as a secret shame, so she saw Clayton as the most perilous so his coming out would be hurtful to her and thus she took it harder than other people.