Yes, but down syndrome usually comes from when someone already has their base sex chromosomes but then one or more extra form. Instead of getting just a xx for female you might get xxx,xxy or even xxyy. With male it will usually be an extra x chromosome attaching to the y, causing the human version of a script overflow. Extra chromosomes do nothing without something to pair with and only cause down syndrome when they attach to sex chromosomes

Down syndrome is actually trisomy 21, which means there is an extra chromosome in the twenty first pair of chromosome. Down syndrome has nothing to do with sex chromosomes. As long as a person has an extra chromosome for the 21st pair regardless of them being male or female, they'll have Down syndrome. Adding an extra X chromosome to a XX female (adding an X to the 23rd pair of chromsomes) results in Turner's Syndrome. XXX results in Triple X syndrome.