Very few outsiders can even create an antimagic field…
Were this player's PC to be using spells like lesser planar binding to bring forth creatures, fretting that the called creatures might use an effect like the spell antimagic field would be the least of his PC's worries.
That is, even were his PC to have done very little research, chances are the called creature simply can't employ an effect like the spell antimagic field. In official, traditional Pathfinder there seems to be only one creature within the binding spells' limits that possesses such an ability: the movanic deva, and it's pretty much just a handsome dude with a sword if it uses its 1/day spell-like ability antimagic field in a room with a low ceiling.
All the other outsiders that can cast or use an ability like the spell antimagic field are well beyond the Hit Dice limit of the typical caster's binding spells: for example, the empyrean angel (HD 25 and CR 20), the demon lord Tsathogga (HD 42 and CR 30), and the Oinodaemon (HD 49 and CR 37) each prepare the spell, and the empyreal lords Black Butterfly (HD 33 and CR 28) and Korda (HD 32 and CR 26) can each 3/day use the spell-like ability. Seriously, the bench just ain't that deep. In short, unless the caster's binding nontraditional outsiders that are themselves casters or one specific kind of angel, this won't be a thing, and it'll never come up.
Nonetheless, this player supposes that a GM could somewhat maliciously decide that his PC calls via a binding spell an outsider that just so happens to possess gear enabling it to use an effect like the spell antimagic field, like the equalizer shield or a scroll of antimagic field. (Note that in this latter case, the GM likely must also customize the creature so that it possesses a sufficient Use Magic Device skill modifier to employ that scroll!)
…And this GM agrees that an outsider can't use antimagic field to free itself from an inward-focused magic circle
This reader agrees with your assessment and would take this even further.
The normal yet inward-focused magic circle against evil description says, "The trapped creature can do nothing that disturbs the circle, directly or indirectly…." That would make this GM rule that an outsider called with a spell like lesser planar binding would be disturbing the circle were it to cast or use an effect like the spell antimagic field. This GM would rule that the magic of the circle spell simply forbids the trapped creature from casting the spell or using the ability just as the magic of the circle forbids the creature from doing anything else that can disturb the circle.
Likewise, because adding special diagrams to an inward-focused magic circle spell means both that "none of its [the trapped creature's] abilities or attacks can cross the diagram" and that the trapped "creature itself cannot disturb the diagram either directly or indirectly, as noted above," the circle effect is then doubly secure against any called creature's antimagic field spell or ability.