Recreating a SSR on a breadboard

Alternatively I would be able to use two mosfets in place of the triac correct?

No for an DC load you only use one FET.
For an AC load you can use two SCRs or one Triac.

A typical FET for this sort of thing is the FDP8880 or the IRLI640 or the IRLB8748 or the BUZ72L.