Won't start one day and fires up first crank next day

Is Ignition box same as ECU, part of the HEI system? If yes it is new
yes ECU is the ignition box (not HEI, but similar)

when you direct wired it to drive it home- how did you do this? Wire from battery to ballast?

Possibly your dual ballast is in backwards...it can work that way, but makes for hard starting- a car i bought recently had the ballast upside down for lack of a better way to say it. There's a U -shape notch in one side to determine how it goes. 67Dart273's test above will sort that out for you....