So I decided to test if runes on heroes that became Zomminions passed their stats to the minions. So I went and repeatedly hit the same enemy (level 40 Scarlet Death in Bone Blight, Chapter 16, Normal Mode) with a caster Zomminion created from a very low level Icepick (but since the minions use Zomm's level 67 instead, it doesn't matter). I started with a runeless Icepick, and saw that the damage dealt was exactly 5420 every time. I then went and gave Icepick a dark Battle rune that gave him a total of +210 attack. When I ran the dungeon again, I saw that he was dealing 4047 damage instead. Confused, I gave Icepick a bulwark rune and removed the battle rune, and he dealt 3278 damage. Remembering i had an attack rune on Zomm that I am too cheap to remove for the sake of a test, I went and ran the test again, but this time with a Solaris becoming a healer minion. For the initial run, I ran the mission with both Zomm and Solaris running 0 additional skill, getting a heal of 3885. I ran the mission again with only Solaris having a focus rune with +122 skill, getting a heal of 4553. Adding a focus rune to Zomm with +306 skill made no difference while Solaris still had the rune equipped, but removing Solaris' weaker focus rune for the next run increased the heal to 4889.
In conclusion, the runes of the hero that became a Zomminion affect the minion, but if that hero had no runes whatsoever, Zomm's runes will apply instead.
Also, I apologize for the block of text, but I wanted to get all my info across to remove any uncertainty in your minds. If I forgot to account for a variable, please let me know, and I will get to doing these tests again properly. And feel free to ask any questions, although I may only check this occasionally, as I am a busy person.