Neons grow too large for a 5 gallon IMO. I'd recommend some of the calmer "nano" fish like ember tetras, Boraras spp, sparkling gouramis, celestial pearl danios, etc.
Healthy neons should easily be able to avoid the filter intake. Did you get them all from the same place? Could be poor quality stock (common with neons). How long has your tank been set up? What is your maintenance routine like? Can you post readings for ammonia, nitrite, and nitrate?