quibblemuch |
No.
Augment Summoning: "Each creature you conjure with any summon spell gains a +4 enhancement bonus to Strength and Constitution for the duration of the spell that summoned it." (bold mine)
The eidolon is not conjured with summon monster or summon nature's ally or any other conjuration (summoning) spell. Therefore, the Augment Summoning feat does not apply.
EDIT: Here, for reference, are some more discussions on the topic.