If I had Mordekaiser, Twice Slain and played it with the Ekko legend ability, can Mordekaiser, Twice Slain save itself?

It depends — Mordekaiser can save itself from that end-of-turn kill, but only if you control another friendly unit on the board to kill as the cost; it can't pay by killing itself.

Boy Who Shattered Time's ability gives the next unit you play "Kill me at end of turn," which is just a delayed kill instruction. Morde's ability is a replacement effect ("if I would die... instead"), so it applies normally here: you kill a friendly unit, and Morde is healed, exhausted, and recalled instead of dying.

What you can't do is name Morde himself as the unit to kill. His own death is the event being replaced, and a replacement effect can only be applied once to an event "or to any Game Actions or events that replace that event" (Rule 370.2) — so killing itself would just leave Morde sitting in the trash with nothing to heal or recall. The kill is a cost, not a target (Rule 355.10.c), so the only requirements are that the chosen unit is a friendly permanent on the board (Rule 429.2.a) and that the once-per-turn limit hasn't already been used this turn.