Growing up in Michigan, Joshua played junior hockey for the Honeybaked U16 midget team and was invited to the USA Hockey Select National Team Evaluation Camp. Following the 2011–12 season, Joshua was drafted by the Sioux Falls Stampede in the third round of the United States Hockey League (USHL). After splitting the 2012–13 season between Honeybaked and the USHL, Joshua was drafted 156th overall by the Plymouth Whalers during the Ontario Hockey League Priority Selection. Despite being drafted, he continued to play in the OHL for the 2013–14 season and was ranked 157th overall amongst North American skaters by the NHL Central Scouting Bureau. Joshua was eventually selected in the fifth round, 128th overall, in the 2014 NHL Entry Draft by the Toronto Maple Leafs. He was unaware he had been drafted until he received a phone call from Toronto management while in the car. After drafting Joshua, Leafs amateur scouting director Dave Morrison spoke highly of him as a player, saying: "He’s tall, very athletic and we saw some flashes of real good skill towards the end of the year. His coaches couldn’t say enough good things about him. He’s very diligent in two ways of the game, a top nine potential player." Joshua spent two full seasons with the Stampede, during which he helped them win the 2015 Clark Cup championship.