Yes
You are reading this correctly. There is no saving throw if the subject is within the up to 20 ft/side area the cage appears to surround. Besides high level magic, shrinking or turning gaseous or whatnot(if the cage version), spells on the caster (if the cage version), you're looking at an hour of sitting still.
Squeezing
what if you arrange your panels so there's a section only 2" wide? Is the creature squeezed between the walls?
First, the space a creature takes up is not the same as the size the creature controls.
A creature's space is the area in feet that it effectively
controls in combat, not an expression of its physical
dimensions. A typical Medium creature isn't 5 feet
wide, for example, but it does control a space that
wide. If a Medium hobgoblin stands in a 5-foot-wide
doorway, other creatures can't get through unless the
hobgoblin lets them. (PHB 192)
The rules on squeezing are pretty clear for most cases:
A creature can squeeze through a space that is large
enough for a creature one size smaller than il. Thus, a
Large creature can squeeze through a passage that's
only 5 feet wide. While squeezing through a space,
a creature must spend 1 extra foot for every foot it
moves there, and it has disadvantage on attack rolls
and Dexterity saving throws. Attack rolls against the
creature have advantage while it's in the smaller space. (PHB 192)
So, a medium creature can squeeze into a space that is Small, which according to DMG page 6 is a 2.5x2.5 ft square it controls.
Squishing
2 feet is less than 2.5 feet so what happens? The rules simply don't say, and a ruling is necessary. So here are some options the DM might rule.
The Creature Takes Bludgeoning Damage
The DM could rule that like in real life when getting stuck in a space that's too small, the creature gets squished into the space, all the things of being squeezed applied, and take some amount of bludgeoning damage. They may also say the space so cramped he's not just losing a foot of movement, but maybe he's in difficult terrain or even restrained.
The Creature is Pushed The Other Way
The DM could rule there isn't enough room and spell isn't a damage spell, and the spell simply shunts them the other way instead.
The Creature is Squeezed
The DM could rule that sizes are an approximation and abstraction. All medium creatures fit inside of a 5' square, but maybe the elf has a slight enough frame can squeeze not only into a 2.5' square but even into a 2' square. The DM could continue that logic to even smaller spaces if they felt appropriate.
Best Answer
An existing forcecage spell's effect is probably unaffected by an incoming antimagic field spell's effect
A spell that creates multiple walls of force—like the the 4th-level Sor/Wiz spell force chest [evoc] (Spell Compendium 97) or the 7th-level Sor/Wiz spell forcecage [evoc] (PH 233)—requires using the rules for the 5th-level Sor/Wiz spell wall of force [evoc] (PH 298-9), and those include how the wall of force spell's effect interacts with the effect of the 6th-level Sor/Wiz spell antimagic field [abjur] (PH 200), this despite the antimagic field spell's description specifying only the spell wall of force.
The spell forcecage says
Compare that with the spell wall of force, which says that it's effect
The reader must refer to the spell wall of force to learn more about the spell forcecage (otherwise, for example, the phrase "vulnerable to a disintegrate spell" is meaningless). And, if trying to determine how, for example, the antimagic field spell's effect interacts with the force cage spell's effect, the reader'll end up back at the spell wall of force anyway.
It would take an extremely strict reading of the spell antimagic field—followed by a personal definition of what a forcecage spell's effect's now-mysterious walls of force were actually made of—to have an existing forcecage spell's effect suppressed by an antimagic field spell's effect.
Walls of force usually block line of effect but an antimagic field spell's effect does not
Normally, the antimagic field spell's effect just doesn't care about a magical effect that would block its line of effect: the antimagic field spell's effect causes the existing magical effect to be suppressed. But, like the spell antimagic field says—and the Rules Compendium confirms—, "A wall of force… isn’t affected by antimagic" (11).
Thus even an existing forcecage spell's barred cage effect allows spells to be cast while those within would be, if uncaged, in an antimagic field spell's effect, although usually not beyond the barred cage effect's walls. While an "otherwise solid barrier with a hole of at least 1 square foot through it does not block a spell’s line of effect" (PH 176), the forcecage spell's barred cage effect has only "half-inch wide [bars] with half-inch gaps between them."
However, casting the spell forcecage while within an existing antimagic field spell's effect and picking as the forcecage spell's effect's point of origin a crosshairs within the antimagic field spell's effect means the forcecage spell's effect is suppressed while the antimagic field spell's effect remains. (However, picking for the forcecage spell's effect's point of origin a crosshairs that's outside the antimagic field spell's effect seems to mean the forcecage spell's effect functions normally. The antimagic field spell's effect doesn't block line of effect (RC 11)—an apparent reversal of how the spell antimagic field functioned according to the Dungeons and Dragons, Third Edition FAQ—, but opinions on exactly how this works vary widely.)