Healing Magic (Cure Light Wounds, Heal, etc) -- Yes, works like normal
Rejuvenate Eidolon (Summoner only spell) -- Yes
Make Whole -- No. Eidolons are outsiders not constructs, even if they look like one.
Natural Healing and rest -- No. Eidolons do not heal with rest as normal creatures do.
I can't find a reference anywhere to ways in which you could heal an Eidolon. Standard healing magic? Make Whole? Anyone? Bueller?
They are living creatures just like any other outsider. Standard healing applies, unless otherwise noted.