This is another utterly crazy claim made by Daniel Cassidy in his trashy book, How The Irish Invented Slang and one that is very easy to disprove. Cassidy claimed that shoo comes from the Irish sitheadh.
Sitheadh is pronounced shee-hoo or shee-ha it is defined by Ó Dónaill as ‘rush, dash, onrush, swoop’. The connection between these meanings and shooing someone or something is not very close and sitheadh is certainly not used when shooing chickens out of the door or telling children to go away. You might use fuisc or amachaigí in cases like this and the action would be described as an ruaig a chur ar dhuine.
So if shoo doesn’t come from Irish, where does it come from? Well, it’s an English word and it’s always been an English word. It is found in English from at least the 15th century and it has a clear cognate in German scheuchen, which means to shoo. Which means that after the fall of the Roman Empire, when the first Saxon-speaking settlers arrived in England from north Germany and Jutland, they used a version of this word to shoo their chickens out of the way, just as their forebears had done in Germany. And in time, this word became shoo in English and scheuchen in German. Where is there any room for Irish influence in this scenario? There isn’t any, of course, because Cassidy was a liar and this is another of his many lies.