Can a naked 4-man trooper squad split its fire? Or does it have to take a heavy weapon or grenades in order to split fire?

When firing with the same weapon all must target the same unit. So, yes, you need a heavy weapon or grenades to split fire.

Either you need a heavy, or the unit needs grenades.
The weapon chosen must hit the same target and cannot splitfire, however if you choose a different weapon they can splitfire. So 4 stormtroopers w/ Impact Grenades can throw 2 grenades at something in R1 while the E-11 blasters fire at something beyond R1 (why you would do this i dunno)

Splitfire rule is mostly reserved for the heavy weapons really, to avoid the old 40k issue of "This weapon is awesome but the target it wants the rest of the unit is useless against...." problems. Dunno how many times in 40k i never used a weapon purely because the unit that takes it wants a completely different target.