Using the feat Expanded Summon Monster, the spell summon monster VII can bring forth a movanic deva that can itself use as a spell-like ability antimagic field. If the deva—or any summoned creature—uses an antimagic field spell or ability, what happens?
Well, let's take the simple approach to that.
Antimagic-field states that:
An antimagic field suppresses any spell or magical effect used within, brought into, or cast into the area, but does not dispel it.
Summoned creatures of any type wink out if they enter an antimagic field. They reappear in the same spot once the field goes away. Time spent winked out counts normally against the duration of the conjuration that is maintaining the creature. If you cast antimagic field in an area occupied by a summoned creature, that has spell resistance, you must make a caster level check against the creature'SR to make it wink out. (The effects of instantaneous conjurations are not affected by an antimagic field because the conjuration itself is no longer in effect, only its result.)
Conjuration being divided into several forms, let's look at the subschools just for the sake of precision:
A calling spell transports a creature from another plane to the plane you are on. The spell grants the creature the one-time ability to return to its plane of origin, although the spell may limit the circumstances under which this is possible. Creatures who are called actually die when they are killed; they do not disappear and reform, as do those brought by a summoning spell (see below). The duration of a calling spell is instantaneous, which means that the called creature can't be dispelled.
Creation : irrelevant
Healing : irrelevant
A summoning spell instantly brings a creature or object to a place you designate. When the spell ends or is dispelled, a summoned creature is instantly sent back to where it came from, [...] When the spell that summoned a creature ends and the creature disappears, all the spells it has cast expire.
A teleportation spell transports one or more creatures or objects a great distance. The most powerful of these spells can cross planar boundaries. Unlike summoning spells, the transportation is (unless otherwise noted) one-way and not dispellable.
Hence, it highly depends how you actually bring the outsider to you. Given how you phrased the question, I suppose you referred to the summoning branch of the Conjuration school.
Any summoned creature placing an antimagic-field on itself would have to succeed an caster level check against its eventual spell resistance to make itself wink out of the plane. If it enters willfully in the field, it just winks out1 (until the antimagic-field expires or the summoning duration runs out, and is handled as normal).
While this is not clearly ruled, I'd argue the creature "winks out of existence" but does not disappear as they would if the summoning expired (since it would reappear on the same spot if the field were to go away and the duration of the summoning continue to run as normal), meaning the spells and SLA they casted would remain until the summoning expires. The creature is therefore not stuck in an infinite loop of contradictions, and is just "not there". Personally, since the creature owns the spell, I'd allow it to release it on their turn, as not to spend the duration of the antimagic-field/summoning (whichever is shorter) blinked out.
Any called creature2 or teleported creature3 would be just fine inside an antimagic field.
1 : I hope something powerful enough to use antimagic-field as an SLA would have SR.
2 : Only way I can think of from the top of my head would be using its true name - can't remember which spell actually call a creature... Planar binding or something?
3 : I'm sure you can figure some shenanigans to teleport an outsider on your plane with Teleportation. Probably involves fetching it yourself.