Does a level 1 summoner's eidolon start with maximum hp just like a PC or is there a rule I missed somewhere?
Most GM's I've seen allow the Eidolon to start with max hp, but, they should actually roll. As in this quote, under Determine Starting Hit Points (from d20pfsrd, not actually in the PRD).
To determine a hit points for levels beyond 1st, roll the dice indicated by its Hit Dice. Creatures whose first Hit Die comes from an NPC class or from his race roll their first Hit Die normally.
1d10 roll + CON or 5 + CON. The last one in case you are playing PFS.
With respect to Pathfinder Society rules, as Eidolons are considered animal companions for most society rules and are an equivalent class feature, you should be following this FAQ (just remember Eidolons have d10 hit die instead of d8):
How do I determine hit points for my animal companion?
Animal companions receive average hit points per hit die. For d8, the average is 4.5. Multiply 4.5 times the number of hit dice your animal companion has and round down. Recalculate hit points for your companion each time it gains additional hit dice. After its hit points are calculated, add in appropriate bonuses from its Constitution modifier, feats, and so on.
Note this refers only to Pathfinder Society but, more focused in your answer, it also implies Eidolons do not get maximum hp at first level, thus, they follow the regular hit points rule:
Hit Points (hp): Hit points are an abstraction signifying how robust and healthy a creature is at the current moment. To determine a creature's hit points, roll the dice indicated by its Hit Dice. A creature gains maximum hit points if its first Hit Die roll is for a character class level. Creatures whose first Hit Die comes from an NPC class or from his race roll their first Hit Die normally.