I think it's debatable whether or not an Eidolon is affected by an Antimagic Field in the first place. First of all, Antimagic Field specifies that it suppresses spells, spell-like abilities, supernatural abilities, and magic items. Summoning an Eidolon isn't specified as ANY of these, only that it's a "ritual" performed by the summoner. Secondly, part of the Eidolon's rules are:
"In addition, due to its tie to its summoner, an eidolon can touch and attack creatures warded by protection from evil and similar effects that prevent contact with summoned creatures."
You could argue that Antimagic Field falls into the category of "similar effects that prevent contact with summoned creatures."
So it's debatable whether or not AMF even affects eidolons. If your GM is set on ruling that it does, the only ways that I know of countering AMF are:
1.) Make your Eidolon too big for AMF (huge or bigger)
2.) Have your Eidolon stay away from the AMF and use only physical/extraordinary ranged attacks against the AMF user
3.) High Spell Resistance
There may be other ways I'm not thinking about. Good luck!