You should not wear running shoes or gym shoes for pickleball. Without proper ankle or heel support, you are at a greater risk for injury.
Tennis and pickleball shoes are specially constructed to protect your ankle and knees from lateral movements and the hard impact that running shoes are not designed for.
Furthermore, tennis and pickleball shoes are made to withstand hard asphalt. A proper Pickleball shoe has a toe guard to protect the shoe against toe dragging.
Cross-trainers are worse because the outsoles have little grips on them that stick to the floor. They can make you trip when you’re backpedaling to retrieve a lob. Stay away from cross-trainers at all costs.