The 45-year-old Welsh actor had the panel completely flummoxed as they were not even sure if the celebrity masquerading as Pig was male or female.
Page told host Joel Dommett: "Oh my goodness, it was really quite funny, I was quite pleased you all thought I was a man."
Jonathan Ross had guessed that Pig was former Top Gear presenter Richard Hammond, while Peter Crouch though she was jockey Frankie Dettori.
Davina McCall thought the dancer was athlete Jessica Ennis-Hill and Oti Mabuse took the country clues to mean she was Countryfile's Anita Rani.
One of Page's clues had been about making a pig's ear of herself.
She confessed: "I made a pig's ear of myself the first time I met Emma Thompson at the Love Actually read-through. I went to the loo and the toilet wouldn't flush and I had to come out and say 'Hello Emma, I've only done a wee!"
Mother-of-four Page said her children had no idea she was taking part in the show and she was so excited to tell them.
She said: "The children don't know! I'm going to go home and say I was Pig!"
The second episode of the new series of the ITV dance show also saw debut dances from Pearly King, Sea Slug, Tomato Sauce, Cactus and Onomatopoeia.
Tomato Sauce wowed everyone with her Irish dancing skills. Her cooking and food clues pointed the judges, and viewers on social media to be convinced she was This Morning chef Clodagh McKenna.
Sea Slug was linked to Gok Wan and Cactus gave strong clues they were a member of boyband Busted.
Guesses for the identities of the other contestants who have danced so far included Kimberly Wyatt or Ashley Roberts for Scissors, Maisie Smith for Odd Socks, and Judge Rob Rinder as Candlestick.
Others thought Pillar and Post, the contest's first duo, could be Rochelle and Marvin Humes or Frankie Poultney and David Seaman.
Watch: Is Amanda Holden Scissors on The Masked Dancer?