While looking for a historical event that happened on my birthday, I came across this on onthisday.com, but I haven't seen this exact date listed anywhere else. Can anyone confirm or deny if this is true?
I can say with confidence that this is not the case. It is fairly well established that the first mass killing at Auschwitz was in late 1941, and June 20, 1944 would be in the middle of the extermination of Hungarian Jews being deported to the camp. I can't find anything which points to that date being particularly significant as part of that pogram. Skimming quickly through Laurence Rees' "Auschwitz: A New History", the only 20 June occurrence he makes specific mention of the date for in relation to the camp would be the 1942 escape by by four Polish prisoners.
The only "hit" that I can find trying to find anything that relates to that date in 1944 was on the USHMM site which makes mention of the Vrba–Wetzler report "going public world wide [...] between June 18 and 22, 1944". But still, June 20 is an awfully specific date. To again borrow from Rees (The Holocaust: A New History), he notes:
The Vrba-Wetzler report circulated in Budapest during May 1944. By late June the news had reached London and by early July the authorities in Washington had been informed.
It would be somewhat important in leading to Admiral Horthy putting a temporary end to the ongoing deportations, but as you can see, there is no particular significance to June 20 there. So this is all to say, I have no idea where that date is coming from here.